{"id":651,"date":"2025-10-09T15:28:10","date_gmt":"2025-10-09T15:28:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=651"},"modified":"2025-12-18T19:32:27","modified_gmt":"2025-12-18T19:32:27","slug":"optimization-and-applications-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/optimization-and-applications-background-youll-need-3\/","title":{"raw":"Optimization and Applications: Background You'll Need 3","rendered":"Optimization and Applications: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Determine whether the Extreme Value Theorem applies to a given function<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"font-claude-response-heading text-text-100 mt-1 -mb-0.5\">The Extreme Value Theorem<\/h2>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">When analyzing functions, we often need to find the largest and smallest values. In multivariable calculus, this becomes crucial for optimization problems involving surfaces and regions in space.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>absolute extrema definitions<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Let [latex]f[\/latex] be a function defined over an interval [latex]I[\/latex] and let [latex]c \\in I[\/latex].<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f[\/latex] has an <strong>absolute maximum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c) \\geq f(x)[\/latex] for all [latex]x \\in I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f[\/latex] has an <strong>absolute minimum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c) \\leq f(x)[\/latex] for all [latex]x \\in I[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Important:<\/strong> The extremum is the [latex]y[\/latex]-value [latex]f(c)[\/latex], while the location is the [latex]x[\/latex]-value [latex]c[\/latex].<\/p>\r\n\r\n<\/section>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Not all functions have absolute extrema. Consider [latex]f(x) = x^3[\/latex] over [latex](-\\infty, \\infty)[\/latex]\u2014as [latex]x[\/latex] increases or decreases without bound, so does [latex]f(x)[\/latex]. This function has neither an absolute maximum nor an absolute minimum.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">However, when certain conditions are met, we can guarantee that both an absolute maximum and absolute minimum exist. The<strong> extreme value theorem<\/strong> guarantees that a continuous function [latex]f[\/latex] over a closed, bounded interval [latex][a,b][\/latex] has both an absolute maximum and an absolute minimum.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>extreme value theorem<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">If [latex]f[\/latex] is <strong>continuous<\/strong> over a <strong>closed, bounded interval<\/strong> [latex][a,b][\/latex], then [latex]f[\/latex] has both:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">An absolute maximum at some point in [latex][a,b][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">An absolute minimum at some point in [latex][a,b][\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">This theorem provides a powerful guarantee, but only when <strong>both conditions<\/strong> are satisfied:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Continuity:<\/strong> The function [latex]f[\/latex] must be continuous over the entire interval [latex][a,b][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Closed, bounded interval:<\/strong> The interval must include both endpoints and have finite length<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">If either condition fails, the theorem does not apply, and the function may or may not have absolute extrema.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\"><strong>Why Both Conditions Matter<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li class=\"font-claude-response-body whitespace-normal break-words\"><strong>Continuity is essential.<\/strong> A single discontinuity can prevent absolute extrema from existing. Consider a piecewise function on [latex][0,4][\/latex] that jumps at [latex]x = 2[\/latex]. Even though the interval is closed and bounded, the discontinuity means the Extreme Value Theorem doesn't apply.<\/li>\r\n \t<li class=\"font-claude-response-body whitespace-normal break-words\"><strong>The interval must be closed and bounded.<\/strong> Open intervals like [latex](a,b)[\/latex] or half-open intervals like [latex][a,b)[\/latex] don't guarantee extrema exist. Similarly, unbounded intervals like [latex][0,\\infty)[\/latex] fail this requirement.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Quick Check for the Extreme Value Theorem:<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Ask yourself two questions:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Is [latex]f[\/latex] continuous on the entire interval [latex][a,b][\/latex]?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Does the interval include both endpoints (closed) and have finite length (bounded)?<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">If yes to both \u2192 EVT applies \u2192 absolute max and min are guaranteed to exist<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">If no to either \u2192 EVT does not apply \u2192 extrema may or may not exist<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Determine whether the Extreme Value Theorem applies to each function. Explain your reasoning.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[latex]f(x) = (x-2)^2[\/latex] on [latex][1, 4][\/latex]<\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[latex]f(x) = \\begin{cases} 2-x^2 &amp; \\text{if } 0 \\leq x &lt; 2 \\ x-3 &amp; \\text{if } 2 \\leq x \\leq 4 \\end{cases}[\/latex] on [latex][0, 4][\/latex]<\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[latex]f(x) = \\frac{x}{2-x}[\/latex] on [latex][0, 2)[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"31512\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"31512\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>The Extreme Value Theorem <strong><strong>applies.\r\n<\/strong><\/strong>\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = (x-2)^2[\/latex] is a polynomial, so it's continuous everywhere, including on [latex][1,4][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The interval [latex][1,4][\/latex] is closed (includes both endpoints) and bounded (finite length)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Both conditions are satisfied, so [latex]f[\/latex] is guaranteed to have an absolute maximum and absolute minimum on [latex][1,4][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The Extreme Value Theorem <strong><strong>does not apply.\r\n<\/strong><\/strong>\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">The interval [latex][0,4][\/latex] is closed and bounded<\/li>\r\n \t<li class=\"whitespace-normal break-words\">However, at [latex]x = 2[\/latex]: [latex]\\lim_{x \\to 2^-} f(x) = 2-2^2 = -2[\/latex] but [latex]f(2) = 2-3 = -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The function is discontinuous at [latex]x = 2[\/latex], so the continuity condition fails<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Without EVT, we cannot guarantee both extrema exist (in this case, the function has an absolute maximum but no absolute minimum)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The Extreme Value Theorem <strong><strong>does not apply.\r\n<\/strong><\/strong>\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = \\frac{x}{2-x}[\/latex] is continuous on [latex][0,2)[\/latex] (the denominator is never zero on this interval)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">However, [latex][0,2)[\/latex] is <strong>not<\/strong> a closed interval\u2014it doesn't include the right endpoint [latex]x = 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The interval condition fails, so EVT doesn't apply<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Without EVT, we cannot guarantee both extrema exist (in this case, the function has an absolute minimum but no absolute maximum)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Common Reasons EVT Fails:<\/strong>\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Discontinuity:<\/strong> Jumps, vertical asymptotes, removable discontinuities, or piecewise functions with mismatched values at transition points<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Open interval:<\/strong> Missing one or both endpoints, like [latex](a,b)[\/latex], [latex](a,b][\/latex], or [latex][a,b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Unbounded interval:<\/strong> Intervals extending to infinity, like [latex][a, \\infty)[\/latex] or [latex](-\\infty, b][\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Always verify <strong>both<\/strong> continuity and the interval type before concluding EVT applies.<\/p>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Determine whether the Extreme Value Theorem applies to a given function<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"font-claude-response-heading text-text-100 mt-1 -mb-0.5\">The Extreme Value Theorem<\/h2>\n<p class=\"font-claude-response-body whitespace-normal break-words\">When analyzing functions, we often need to find the largest and smallest values. In multivariable calculus, this becomes crucial for optimization problems involving surfaces and regions in space.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>absolute extrema definitions<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Let [latex]f[\/latex] be a function defined over an interval [latex]I[\/latex] and let [latex]c \\in I[\/latex].<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]f[\/latex] has an <strong>absolute maximum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c) \\geq f(x)[\/latex] for all [latex]x \\in I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f[\/latex] has an <strong>absolute minimum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c) \\leq f(x)[\/latex] for all [latex]x \\in I[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Important:<\/strong> The extremum is the [latex]y[\/latex]-value [latex]f(c)[\/latex], while the location is the [latex]x[\/latex]-value [latex]c[\/latex].<\/p>\n<\/section>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Not all functions have absolute extrema. Consider [latex]f(x) = x^3[\/latex] over [latex](-\\infty, \\infty)[\/latex]\u2014as [latex]x[\/latex] increases or decreases without bound, so does [latex]f(x)[\/latex]. This function has neither an absolute maximum nor an absolute minimum.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">However, when certain conditions are met, we can guarantee that both an absolute maximum and absolute minimum exist. The<strong> extreme value theorem<\/strong> guarantees that a continuous function [latex]f[\/latex] over a closed, bounded interval [latex][a,b][\/latex] has both an absolute maximum and an absolute minimum.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>extreme value theorem<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">If [latex]f[\/latex] is <strong>continuous<\/strong> over a <strong>closed, bounded interval<\/strong> [latex][a,b][\/latex], then [latex]f[\/latex] has both:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">An absolute maximum at some point in [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">An absolute minimum at some point in [latex][a,b][\/latex]<\/li>\n<\/ul>\n<\/section>\n<p class=\"font-claude-response-body whitespace-normal break-words\">This theorem provides a powerful guarantee, but only when <strong>both conditions<\/strong> are satisfied:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Continuity:<\/strong> The function [latex]f[\/latex] must be continuous over the entire interval [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Closed, bounded interval:<\/strong> The interval must include both endpoints and have finite length<\/li>\n<\/ol>\n<p class=\"font-claude-response-body whitespace-normal break-words\">If either condition fails, the theorem does not apply, and the function may or may not have absolute extrema.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\"><strong>Why Both Conditions Matter<\/strong><\/p>\n<ul>\n<li class=\"font-claude-response-body whitespace-normal break-words\"><strong>Continuity is essential.<\/strong> A single discontinuity can prevent absolute extrema from existing. Consider a piecewise function on [latex][0,4][\/latex] that jumps at [latex]x = 2[\/latex]. Even though the interval is closed and bounded, the discontinuity means the Extreme Value Theorem doesn&#8217;t apply.<\/li>\n<li class=\"font-claude-response-body whitespace-normal break-words\"><strong>The interval must be closed and bounded.<\/strong> Open intervals like [latex](a,b)[\/latex] or half-open intervals like [latex][a,b)[\/latex] don&#8217;t guarantee extrema exist. Similarly, unbounded intervals like [latex][0,\\infty)[\/latex] fail this requirement.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Quick Check for the Extreme Value Theorem:<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Ask yourself two questions:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Is [latex]f[\/latex] continuous on the entire interval [latex][a,b][\/latex]?<\/li>\n<li class=\"whitespace-normal break-words\">Does the interval include both endpoints (closed) and have finite length (bounded)?<\/li>\n<\/ol>\n<p class=\"font-claude-response-body whitespace-normal break-words\">If yes to both \u2192 EVT applies \u2192 absolute max and min are guaranteed to exist<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">If no to either \u2192 EVT does not apply \u2192 extrema may or may not exist<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Determine whether the Extreme Value Theorem applies to each function. Explain your reasoning.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\n<p class=\"font-claude-response-body whitespace-normal break-words\">[latex]f(x) = (x-2)^2[\/latex] on [latex][1, 4][\/latex]<\/p>\n<\/li>\n<li>\n<p class=\"font-claude-response-body whitespace-normal break-words\">[latex]f(x) = \\begin{cases} 2-x^2 & \\text{if } 0 \\leq x < 2 \\ x-3 & \\text{if } 2 \\leq x \\leq 4 \\end{cases}[\/latex] on [latex][0, 4][\/latex]<\/p>\n<\/li>\n<li>\n<p class=\"font-claude-response-body whitespace-normal break-words\">[latex]f(x) = \\frac{x}{2-x}[\/latex] on [latex][0, 2)[\/latex]<\/p>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q31512\">Show Answer<\/button><\/p>\n<div id=\"q31512\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>The Extreme Value Theorem <strong><strong>applies.<br \/>\n<\/strong><\/strong><\/p>\n<ul class=\"&#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = (x-2)^2[\/latex] is a polynomial, so it&#8217;s continuous everywhere, including on [latex][1,4][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The interval [latex][1,4][\/latex] is closed (includes both endpoints) and bounded (finite length)<\/li>\n<li class=\"whitespace-normal break-words\">Both conditions are satisfied, so [latex]f[\/latex] is guaranteed to have an absolute maximum and absolute minimum on [latex][1,4][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The Extreme Value Theorem <strong><strong>does not apply.<br \/>\n<\/strong><\/strong><\/p>\n<ul class=\"&#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">The interval [latex][0,4][\/latex] is closed and bounded<\/li>\n<li class=\"whitespace-normal break-words\">However, at [latex]x = 2[\/latex]: [latex]\\lim_{x \\to 2^-} f(x) = 2-2^2 = -2[\/latex] but [latex]f(2) = 2-3 = -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The function is discontinuous at [latex]x = 2[\/latex], so the continuity condition fails<\/li>\n<li class=\"whitespace-normal break-words\">Without EVT, we cannot guarantee both extrema exist (in this case, the function has an absolute maximum but no absolute minimum)<\/li>\n<\/ul>\n<\/li>\n<li>The Extreme Value Theorem <strong><strong>does not apply.<br \/>\n<\/strong><\/strong><\/p>\n<ul class=\"&#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = \\frac{x}{2-x}[\/latex] is continuous on [latex][0,2)[\/latex] (the denominator is never zero on this interval)<\/li>\n<li class=\"whitespace-normal break-words\">However, [latex][0,2)[\/latex] is <strong>not<\/strong> a closed interval\u2014it doesn&#8217;t include the right endpoint [latex]x = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The interval condition fails, so EVT doesn&#8217;t apply<\/li>\n<li class=\"whitespace-normal break-words\">Without EVT, we cannot guarantee both extrema exist (in this case, the function has an absolute minimum but no absolute maximum)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Common Reasons EVT Fails:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Discontinuity:<\/strong> Jumps, vertical asymptotes, removable discontinuities, or piecewise functions with mismatched values at transition points<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Open interval:<\/strong> Missing one or both endpoints, like [latex](a,b)[\/latex], [latex](a,b][\/latex], or [latex][a,b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Unbounded interval:<\/strong> Intervals extending to infinity, like [latex][a, \\infty)[\/latex] or [latex](-\\infty, b][\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Always verify <strong>both<\/strong> continuity and the interval type before concluding EVT 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