Derivatives and the Shape of a Graph: Fresh Take

  • Use the first derivative to determine where a function is going up or down, and identify points that might be local highs or lows
  • Apply the second derivative to find out where a function curves upward or downward and locate points where this curvature changes
  • Use the second derivative test to determine if a point on a graph is the highest or lowest within specific sections

The First Derivative Test

The Main Idea 

  • Used to determine local extrema of a continuous function
  • Theoretical Basis:
    • A function changes from increasing to decreasing (or vice versa) at local extrema
    • This change occurs at critical points where [latex]f'(x) = 0[/latex] or [latex]f'(x)[/latex] is undefined
  • Test Conditions:
    • [latex]f(x)[/latex] is continuous over an interval containing the critical point [latex]c[/latex]
    • [latex]f(x)[/latex] is differentiable over the interval, except possibly at [latex]c[/latex]
  • Key Results:
    • Local maximum: [latex]f'(x)[/latex] changes from positive to negative at [latex]c[/latex]
    • Local minimum: [latex]f'(x)[/latex] changes from negative to positive at [latex]c[/latex]
    • Neither: [latex]f'(x)[/latex] doesn’t change sign at [latex]c[/latex]

How to Apply the First Derivative Test

  1. Find critical points: Solve [latex]f'(x) = 0[/latex] and where [latex]f'(x)[/latex] is undefined
  2. Divide the domain into intervals using the critical points
  3. Determine the sign of [latex]f'(x)[/latex] in each interval
  4. Analyze sign changes to identify local extrema

Use the first derivative test to locate all local extrema for [latex]f(x)=−x^3+\frac{3}{2}x^2+18x[/latex].

Use the first derivative test to find all local extrema for [latex]f(x)=\sqrt[3]{x-1}[/latex].

Concavity and Points of Inflection

The Main Idea 

  • Concavity:
    • Concave Up: [latex]f'(x)[/latex] is increasing; [latex]f''(x) > 0[/latex]
    • Concave Down: [latex]f'(x)[/latex] is decreasing; [latex]f''(x) < 0[/latex]
  • Concavity Test: For a twice-differentiable function [latex]f(x)[/latex] on interval [latex]I[/latex]:
    • If [latex]f''(x) > 0[/latex] for all [latex]x \in I[/latex], [latex]f[/latex] is concave up on [latex]I[/latex]
    • If [latex]f''(x) < 0[/latex] for all [latex]x \in I[/latex], [latex]f[/latex] is concave down on [latex]I[/latex]
  • Inflection Points:
    • Points where concavity changes
    • Occur where [latex]f''(x) = 0[/latex] or [latex]f''(x)[/latex] is undefined
    • [latex]f(x)[/latex] must be continuous at these points
  • Relationship to Derivatives:
    • First derivative ([latex]f'[/latex]) indicates increasing/decreasing
    • Second derivative ([latex]f''[/latex]) indicates concavity

How to Analyze Concavity and Find Inflection Points

  1. Find [latex]f''(x)[/latex]
  2. Determine where [latex]f''(x) = 0[/latex] or is undefined
  3. Use these points to divide the domain into intervals
  4. Test the sign of [latex]f''(x)[/latex] in each interval
  5. Identify where concavity changes to locate inflection points

For [latex]f(x)=−x^3+\frac{3}{2}x^2+18x[/latex], find all intervals where [latex]f[/latex] is concave up and all intervals where [latex]f[/latex] is concave down.

Analyze the concavity and find inflection points of [latex]f(x) = x^3 - 3x^2 + 1[/latex].

The Second Derivative Test

The Main Idea 

  1. Purpose:
    • Used to determine the nature of critical points (local maxima or minima)
  2. Theorem Statement: For a twice-differentiable function [latex]f(x)[/latex] with [latex]f'(c) = 0[/latex]:
    • If [latex]f''(c) > 0[/latex], [latex]f[/latex] has a local minimum at [latex]c[/latex]
    • If [latex]f''(c) < 0[/latex], [latex]f[/latex] has a local maximum at [latex]c[/latex]
    • If [latex]f''(c) = 0[/latex], the test is inconclusive
  3. Relationship to Concavity:
    • [latex]f''(c) > 0[/latex] indicates concave up at [latex]c[/latex]
    • [latex]f''(c) < 0[/latex] indicates concave down at [latex]c[/latex]
  4. Advantages:
    • Often simpler than the First Derivative Test
    • Provides information about concavity
  5. Limitations:
    • Requires [latex]f''(x)[/latex] to be continuous near [latex]c[/latex]
    • Inconclusive when [latex]f''(c) = 0[/latex]

How to Apply the Second Derivative Test

  1. Find critical points by solving [latex]f'(x) = 0[/latex]
  2. Calculate [latex]f''(x)[/latex]
  3. Evaluate [latex]f''(x)[/latex] at each critical point
  4. Use the test to classify each critical point
  5. For inconclusive cases, use the First Derivative Test

Consider the function [latex]f(x)=x^3-\left(\frac{3}{2}\right)x^2-18x[/latex]. The points [latex]c=3,-2[/latex] satisfy [latex]f^{\prime}(c)=0[/latex]. Use the second derivative test to determine whether [latex]f[/latex] has a local maximum or local minimum at those points.

Find and classify the local extrema of [latex]f(x) = x^4 - 4x^3 + 2[/latex] using the Second Derivative Test.