Section 4.1 Notes - Maximum and Minimum Values

Maximum and Minimum Values

Let $c$ be a number in the domain $D$ of a function $f$.

• Absolute Maximum Value: $f(c)$ is the absolute maximum value of $f$ on $D$ if $f(c) \geq f(x)$ for all $x$ in $D$. ($\forall x \in D$)

• Absolute Minimum Value: $f(c)$ is the absolute minimum value of $f$ on $D$ if $f(c) \leq f(x)$ for all $x$ in $D$.

Local/Relative Max and Min

• $f(c)$ is a local/relative max of $f$ if f(c) > f(x) when $x$ is near $c$.

• $f(c)$ is a local/relative min of $f$ if f(c) < f(x) when $x$ is near $c$.

Fermat's Theorem

If $f$ has a local max or min at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.

Critical Number/Value

A critical number/value of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist (dne).

• If $c$ has a max or min, it is a critical value.

• If $c$ is a critical value, it MAY be a max or min.

The Extreme Value Theorem

If $f$ is continuous on a closed interval $[a, b]$, then $f$ attains an absolute max value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $[a, b]$.

Finding Max and Min

1. Find the critical values.

2. Evaluate $f(a)$ and $f(b)$.

3. The largest y-coordinate is the max, and the smallest is the min.

Example:

$f(x) = 3x^4 - 4x^3 - 12x^2 + 1$ in $[-2, 3]$

1. Find critical values in $[-2, 3]$.
   $f'(x) = 12x^3 - 12x^2 - 24x = 12x(x^2 - x - 2) = 12x(x - 2)(x + 1) = 0$
   Critical values: $x = 0, 2, -1$

2. Evaluate at critical values and endpoints:

   • $f(0) = 1$

   • $f(2) = -31$

   • $f(-1) = -4$

   • $f(-2) = 33$

   • $f(3) = -8$

Absolute max of 33 at $x = -2$
Absolute min of -31 at $x = 2$