Extrema, The Mean Value Theorem, and Monotonicity in Calculus

Extreme Values and Critical Points

Definition of Critical Points

Critical points are essential in analyzing the behavior of functions, especially when searching for local and absolute extrema.

• Critical Point (or Critical Number): A number c in the domain of f is a critical point of f if either:

  • $f'(c) = 0$

  • or $f'(c)$ does not exist

• Critical points are candidates for local maxima, minima, or points of inflection.

Example: For $f(x) = x^4 - 4x^3$, the critical points are found by solving $f'(x) = 0$.

Finding Critical Points: Examples

• Example 1: $f(x) = 18 \cos x + 9 \sin^2 x$ on $[0, 2\pi]$

  • Compute $f'(x)$ and solve $f'(x) = 0$ to find critical points within the interval.

• Example 2: $f(x) = 1 - (x-1)^{2/3}$

  • Check where $f'(x) = 0$ or $f'(x)$ does not exist. For this function, $x=1$ is a critical point because the derivative does not exist there.

Absolute Extrema on a Closed Interval

The Extreme Value Theorem

If f is continuous on a closed interval $[a, b]$, then f attains both an absolute maximum and an absolute minimum value, each at either a critical point or at an endpoint.

• Key Hypotheses:

  • f must be continuous on $[a, b]$

  • The interval must be closed

• If these conditions fail, absolute extrema are not guaranteed.

The Closed Interval Method

To find the absolute maximum and minimum of a continuous function on $[a, b]$:

1. Verify that f is continuous on $[a, b]$ and the interval is closed.

2. Find all critical points of f in $(a, b)$.

3. Evaluate f at each critical point and at the endpoints $a$ and $b$.

4. Compare all values: the largest is the absolute maximum, the smallest is the absolute minimum.

Example: For $f(x) = x^3 - 3x^2 - 9x + 5$ on $[-2, 4]$, compute $f$ at all critical points and endpoints, then compare.

Rolle’s Theorem and the Mean Value Theorem (MVT)

Rolle’s Theorem

Rolle’s Theorem is a special case of the Mean Value Theorem, providing a guarantee about horizontal tangents.

• If f is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists $c$ in $(a, b)$ such that $f'(c) = 0$.

Example: For $f(x) = 5 - 12x + 3x^2$ on $[1, 3]$, check the hypotheses and solve $f'(c) = 0$.

The Mean Value Theorem (MVT)

The MVT generalizes Rolle’s Theorem to cases where $f(a) \neq f(b)$.

• If f is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b - a}$

• The instantaneous rate of change at $c$ equals the average rate of change over $[a, b]$.

• Rolle’s Theorem is the special case where $f(a) = f(b)$, so the right side is zero.

Example: For $f(x) = 3x^2 + 2x + 5$ on $[-1, 1]$, find $c$ such that $f'(c)$ equals the average rate of change.

Monotonicity: Increasing and Decreasing Functions

Definitions

• Increasing: $f$ is increasing on $(a, b)$ if $f(x_1) < f(x_2)$ for all $x_1 < x_2$ in $(a, b)$.

• Decreasing: $f$ is decreasing on $(a, b)$ if $f(x_1) > f(x_2)$ for all $x_1 < x_2$ in $(a, b)$.

• Monotonic: $f$ is monotonic if it is either increasing or decreasing on the interval.

Increasing/Decreasing Test

• If $f'(x) > 0$ for all $x$ in $(a, b)$, then $f$ is increasing on $(a, b)$.

• If $f'(x) < 0$ for all $x$ in $(a, b)$, then $f$ is decreasing on $(a, b)$.

The First Derivative Test

• Let $c$ be a critical point of a continuous function $f$.

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

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

• If $f'$ does not change sign at $c$, then $f(c)$ is neither a local max nor min.

Recipe for Analyzing Monotonicity and Local Extrema:

1. Find the domain and critical points of $f$.

2. Split the domain at critical points and points where $f$ is undefined.

3. Test the sign of $f'$ in each interval (sign chart).

4. Determine where $f$ increases/decreases and apply the First Derivative Test.

Examples

• Example 1: $f(x) = 10x^3 - 6x^5$

  • Find $f'(x)$, set to zero, and analyze sign changes to determine intervals of increase/decrease and local extrema.

• Example 2: $f(x) = x e^x$

  • Use the product rule to find $f'(x)$, solve $f'(x) = 0$, and analyze intervals.

Concavity and Inflection Points

Concavity

• Concave Up: $f$ is concave up on $(a, b)$ if $f'$ is increasing (i.e., $f'' > 0$).

• Concave Down: $f$ is concave down on $(a, b)$ if $f'$ is decreasing (i.e., $f'' < 0$).

Concavity Test

• If $f''(x) > 0$ for all $x$ in $(a, b)$, then $f$ is concave up on $(a, b)$.

• If $f''(x) < 0$ for all $x$ in $(a, b)$, then $f$ is concave down on $(a, b)$.

Inflection Points

• Inflection Point: A point $P = (c, f(c))$ is an inflection point if $f$ is continuous at $c$ and $f$ changes concavity at $x = c$ (i.e., $f''$ changes sign).

• Note: $f''(c) = 0$ is necessary but not sufficient for an inflection point; the sign of $f''$ must change.

Example: For $f(x) = x^4$, $f''(0) = 0$ but $f''$ does not change sign at $x=0$, so $(0,0)$ is not an inflection point.

The Second Derivative Test

• If $c$ is a critical point with $f'(c) = 0$ and $f''(c)$ exists:

  • If $f''(c) > 0$, $f(c)$ is a local minimum (concave up).

  • If $f''(c) < 0$, $f(c)$ is a local maximum (concave down).

  • If $f''(c) = 0$, the test is inconclusive; use the First Derivative Test.

Summary Table: Tests for Extrema and Concavity

Test	What to Check	Conclusion
First Derivative Test	Sign change of $f'$ at critical point	+ to −: local max; − to +: local min; no sign change: neither
Second Derivative Test	$f'(c) = 0$, $f''(c)$ exists	$f''(c) > 0$: local min; $f''(c) < 0$: local max; $f''(c) = 0$: inconclusive
Concavity Test	Sign of $f''$ on interval	$f'' > 0$: concave up; $f'' < 0$: concave down
Inflection Point	$f''(c) = 0$ and $f''$ changes sign at $c$	$(c, f(c))$ is an inflection point

Practice and Applications

• Use the closed interval method to find absolute extrema.

• Apply the MVT to real-world problems (e.g., average speed guarantees instantaneous speed).

• Analyze monotonicity and concavity to sketch graphs and solve optimization problems.

Key Formulas

• Critical Points: $f'(x) = 0$ or $f'(x)$ DNE

• MVT: $f'(c) = \frac{f(b) - f(a)}{b - a}$

• First Derivative Test: Analyze sign changes of $f'$

• Second Derivative Test: $f''(c) > 0$ (min), $f''(c) < 0$ (max)

• Concavity: $f''(x) > 0$ (up), $f''(x) < 0$ (down)

Additional info: These notes include both theoretical results and practical methods for finding and classifying extrema, as well as for analyzing the shape of graphs using derivatives. The content is suitable for Calculus I students preparing for exams on these topics.