What Derivatives Tell Us: 4.3

What Derivatives Tell Us

Introduction

This section explores how the first and second derivatives of a function provide critical information about its behavior, including intervals of increase and decrease, local and absolute extrema, concavity, and inflection points. These concepts are foundational for analyzing and graphing functions in calculus.

Intervals of Increase and Decrease

Critical Points and Monotonicity

• Critical Point: A value x = c where $f'(c) = 0$ or $f'(c)$ is undefined.

• Increasing Interval: $f'(x) > 0$ on an interval implies f is increasing there.

• Decreasing Interval: $f'(x) < 0$ on an interval implies f is decreasing there.

Example 1: Find where $f(x) = -10 - x^2$ is increasing or decreasing.

• $f'(x) = -2x$

• Set $f'(x) = 0 \implies x = 0$

• Test intervals: $(-\infty, 0)$ (increasing), $(0, \infty)$ (decreasing)

Example 2: $f(x) = -x^3 - 5x$

• $f'(x) = -3x^2 - 5$

• Set $f'(x) = 0 \implies x^2 = -5/3$ (no real solution)

• Since $f'(x) < 0$ for all $x$, $f$ is always decreasing on $(-\infty, \infty)$

Identifying Local Extrema Using the First Derivative Test

First Derivative Test

• If $f'(x)$ changes from positive to negative at $x = c$, then $f$ has a local maximum at $c$.

• If $f'(x)$ changes from negative to positive at $x = c$, then $f$ has a local minimum at $c$.

• If $f'(x)$ does not change sign at $c$, then $c$ is not a local extremum.

Example: Find local extrema for $f(x) = 2x^3 + 3x^2 - 120x + 3$ on $[-5, 7]$.

• $f'(x) = 6x^2 + 6x - 120 = 6(x + 5)(x - 4)$

• Critical points: $x = -5$ (endpoint), $x = 4$ (critical point)

• Test intervals: $f'(x) < 0$ on $(-5, 4)$, $f'(x) > 0$ on $(4, 7)$

• Conclusion: Local minimum at $f(4) = -301$

Absolute Extrema

Finding Absolute Maximum and Minimum

• Evaluate $f(x)$ at all critical points and endpoints of the interval.

• The largest value is the absolute maximum; the smallest is the absolute minimum.

Example: $f(x) = 4x^5 + 5x^4 - 40x^3 + 4$ on $[-4, 3]$

• $f'(x) = 20x^4 + 20x^3 - 120x^2 = 20x^2(x + 3)(x - 2)$

• Critical points: $x = -3, 0, 2$

• Evaluate $f(x)$ at $x = -4, -3, 0, 2, 3$

• Absolute max: $f(-3) = 517$

• Absolute min: $f(-4) = -252$

Intervals of Concavity

Concavity and the Second Derivative

• If $f''(x) > 0$ on an interval, $f$ is concave up there.

• If $f''(x) < 0$ on an interval, $f$ is concave down there.

Inflection Points

Where Concavity Changes

• An inflection point occurs at $x = c$ if $f''(c) = 0$ or $f''(c)$ is undefined and $f''(x)$ changes sign at $c$.

• At inflection points, the graph of $f$ changes from concave up to concave down or vice versa.

Example: $f(x) = -x^6 + 6x^5 - 6x + 5$

• $f''(x) = -30x^4 + 120x^3 = -30x^3(x - 4)$

• Set $f''(x) = 0 \implies x = 0, 4$

• Intervals: Concave up on $(0, 4)$, concave down on $(-\infty, 0)$ and $(4, \infty)$

• Inflection points at $x = 0, 4$

Identifying Local Extrema Using the Second Derivative Test

Second Derivative Test

• Suppose $c$ is a critical point with $f'(c) = 0$:

  • If $f''(c) > 0$, $f$ has a local minimum at $c$.

  • If $f''(c) < 0$, $f$ has a local maximum at $c$.

  • If $f''(c) = 0$, the test is inconclusive; use the first derivative test.

Summary Table: Tests for Extrema and Concavity

Test	Condition	Conclusion
First Derivative Test	$f'(x)$ changes sign at $c$	Local max or min at $c$
Second Derivative Test	$f'(c) = 0$, $f''(c) > 0$	Local min at $c$
Second Derivative Test	$f'(c) = 0$, $f''(c) < 0$	Local max at $c$
Concavity	$f''(x) > 0$	Concave up
Concavity	$f''(x) < 0$	Concave down
Inflection Point	$f''(c) = 0$ and $f''(x)$ changes sign at $c$	Inflection point at $c$

Additional info: The notes include graphical illustrations and number line diagrams to support the analysis of intervals and extrema. All examples are standard for Calculus I and reinforce the application of derivative tests.