Maxima and Minima: 4.1

Maxima and Minima

Introduction

This section explores the concepts of maxima and minima in calculus, focusing on how to identify and classify extreme values of functions. These ideas are fundamental for understanding optimization problems and the behavior of functions on closed intervals.

Extreme Value Theorem

Statement and Implications

• Extreme Value Theorem: If a function f is continuous on a closed interval $[a, b]$, then f attains both an absolute maximum and an absolute minimum on that interval.

• Absolute maximum: The largest value of f on $[a, b]$; occurs at some point $c$ where $f(c) \geq f(x)$ for all $x$ in $[a, b]$.

• Absolute minimum: The smallest value of f on $[a, b]$; occurs at some point $d$ where $f(d) \leq f(x)$ for all $x$ in $[a, b]$.

• Local (relative) maximum: $f(c)$ is a local maximum if $f(c) \geq f(x)$ for all $x$ near $c$.

• Local (relative) minimum: $f(c)$ is a local minimum if $f(c) \leq f(x)$ for all $x$ near $c$.

Example (from graph):

• Absolute maximum: $f(c_3)$

• Absolute minimum: $f(a)$

• Local maxima: $f(c_1)$, $f(c_3)$

• Local minimum: $f(c_2)$

Critical Points and Local Extrema

Definition and Identification

• Critical point: A point $c$ in the domain of $f$ where either $f'(c) = 0$ or $f'(c)$ does not exist (DNE).

• Local extreme values (maxima or minima) can only occur at critical points or endpoints of the interval.

Types of critical points:

• $f'(c) = 0$ (horizontal tangent): e.g., $c_1$, $c_2$ in the graph.

• $f'(c)$ DNE (does not exist): e.g., $c_3$ in the graph.

Important Note

• Not all critical points are local extrema. Some critical points may not correspond to a local maximum or minimum (e.g., inflection points or cusps).

Examples:

• At $f'(c) = 0$, the function may have a horizontal inflection point, not a max/min.

• At $f'(c)$ DNE, the function may have a cusp or vertical tangent, not necessarily a max/min.

Finding Critical Points: Examples

Example 1: Polynomial Function

• Given $f(x) = 3x^2 + 5x - 2$

• Compute derivative: $f'(x) = 6x + 5$

• Set $f'(x) = 0$: $6x + 5 = 0 \implies x = -\frac{5}{6}$ (critical point)

• $f'(x)$ DNE: Not applicable for polynomials

Example 2: Root Function on $[0,1]$

• Given $f(x) = \sqrt{x}$

• Compute derivative: $f'(x) = \frac{1}{2\sqrt{x}}$

• Set $f'(x) = 0$: No solution (denominator never zero)

• $f'(x)$ DNE at $x = 0$ (endpoint)

Example 3: Rational Function

• Given $f(x) = \frac{16x}{x^2 + 64}$

• Compute derivative using quotient rule: $f'(x) = \frac{16(x^2 + 64) - 16x \cdot 2x}{(x^2 + 64)^2}$ $= \frac{16(x^2 + 64) - 32x^2}{(x^2 + 64)^2}$ $= \frac{16x^2 + 1024 - 32x^2}{(x^2 + 64)^2}$ $= \frac{-16x^2 + 1024}{(x^2 + 64)^2}$

• Set numerator to zero: $-16x^2 + 1024 = 0 \implies x^2 = 64 \implies x = \pm 8$ (critical points)

• $f'(x)$ DNE: Not applicable (denominator never zero for real $x$)

Finding Absolute Extreme Values on a Closed Interval

Procedure

1. Locate all critical points $c_1, c_2, \ldots, c_n$ in the open interval $(a, b)$.

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

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

Example 1: Cubic Polynomial on $\left[-\frac{1}{4}, 4\right]$

• $f(x) = x^3 - 3x^2 + 1$

• Find critical points: $f'(x) = 3x^2 - 6x = 3x(x-2) = 0 \implies x = 0, 2$

• Evaluate:

  • $f(0) = 1$

  • $f(2) = -3$

  • $f(-\frac{1}{4}) = -\frac{1}{64} - 3\left(\frac{1}{16}\right) + 1 =$ (calculation omitted for brevity)

  • $f(4) = 17$

• Absolute maximum: $f(4) = 17$

• Absolute minimum: $f(2) = -3$

Example 2: Quartic Polynomial on $[0,2]$

• $f(x) = 2x^3 + 3x^2 - 12x + 4$

• Find critical points: $f'(x) = 6x^2 + 6x - 12 = 6(x+2)(x-1) = 0 \implies x = 1, -2$

• Only $x=1$ is in $[0,2]$

• Evaluate:

  • $f(1) = -3$

  • $f(0) = 4$

  • $f(2) = 8$

• Absolute maximum: $f(2) = 8$

• Absolute minimum: $f(1) = -3$

Example 3: Radical Function on $[-7,9]$

• $h(x) = (x-1)^{1/3}$

• Find critical points: $h'(x) = \frac{1}{3}(x-1)^{-2/3}$

• $h'(x)$ DNE at $x=1$ (critical point)

• Evaluate:

  • $h(1) = 0$

  • $h(-7) = (-8)^{1/3} = -2$

  • $h(9) = (8)^{1/3} = 2$

• Absolute maximum: $h(9) = 2$

• Absolute minimum: $h(-7) = -2$

Summary Table: Types of Extreme Values

Type	Definition	Where it Occurs
Absolute Maximum	Largest value of $f$ on $[a, b]$	Critical point or endpoint
Absolute Minimum	Smallest value of $f$ on $[a, b]$	Critical point or endpoint
Local Maximum	$f(c) \geq f(x)$ for $x$ near $c$	Critical point
Local Minimum	$f(c) \leq f(x)$ for $x$ near $c$	Critical point

Key Takeaways

• Extreme values (maxima and minima) are found at critical points and endpoints of closed intervals.

• Critical points occur where $f'(x) = 0$ or $f'(x)$ does not exist.

• Not all critical points are local extrema; further analysis is needed.

• To find absolute extrema on $[a, b]$, evaluate $f$ at all critical points in $(a, b)$ and at $a$, $b$.