if $f$ has an absolute maximum or absolute minimum at $c,$ we say $f$ has an absolute extremum at $c$

if $f (c) \ge f (x)$ for all $x$ in the domain of $f,$ we say $f$ has an absolute maximum at $c$

if $f (c) \le f (x)$ for all $x$ in the domain of $f,$ we say $f$ has an absolute minimum at $c$

the point $(c \text{,} f (c))$ a critical point of $f$

if $f ' (c) = 0$ or $f ' (c)$ is undefined, we say that $c$ is a critical number of $f$

extreme value theorem

if $f$ is a continuous function over a finite, closed interval, then $f$ has an absolute maximum and an absolute minimum

if $f$ has a local extremum at $c,$ then $c$ is a critical point of $f$

if $f$ has a local maximum or local minimum at $c,$ we say $f$ has a local extremum at $c$

if there exists an interval $I$ such that $f (c) \ge f (x)$ for all $x \in I,$ we say $f$ has a local maximum at $c$

if there exists an interval $I$ such that $f (c) \le f (x)$ for all $x \in I,$ we say $f$ has a local minimum at $c$

4.3 Maxima and Minima: practice

If you are finding an absolute minimum over an interval $[a, b],$ why do you need to check the endpoints? Draw a graph that supports your hypothesis.

When you are checking for critical points, explain why you also need to determine points where $f^{'} (x)$ is undefined. Draw a graph to support your explanation.

If you are finding an absolute minimum over an interval $[a, b],$ why do you need to check the endpoints? Draw a graph that supports your hypothesis.

When you are checking for critical points, explain why you also need to determine points where $f^{'} (x)$ is undefined. Draw a graph to support your explanation.

Can you have a finite absolute maximum for $y = a x^{3} + b x^{2} + c x + d$ over $(- \infty, \infty)$ assuming a is non-zero? Explain why or why not using graphical arguments.

Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.

[T] Graph the function $y = e^{a x} .$ For which values of $a,$ on any infinite domain, will you have an absolute minimum and absolute maximum?

For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume the graph represents the entirety of each function. For any extrema located at an endpoint, approximate the x-value.

The function graphed starts at (−2.2, 10), decreases rapidly to (−2, −11), increases to (−1, 5) before decreasing slowly to (1, 3), at which point it increases to (2, 7), and then decreases to (3, −20).

The function graphed starts at (−2.5, 1), decreases rapidly to (−2, −1.25), increases to (−1, 0.25) before decreasing slowly to (0, 0.2), at which point it increases slowly to (1, 0.25), then decreases rapidly to (2, −1.25), and finally increases to (2.5, 1).

For the following problems, draw graphs of $f (x),$ which is continuous, over the interval $[- 4, 4]$ with the following properties:

Absolute minimum at $x = 1$ and absolute maximum at $x = 2$

For the following problems, draw graphs of $f (x),$ which is continuous, over the interval $[- 4, 4]$ with the following properties:

Absolute maxima at $x = 2$ and $x = - 3,$ local minimum at $x = 1,$ and absolute minimum at $x = 4$

For the following exercises, find the critical numbers in the domains of the following functions.

$y = 4 x - x^{2}$

For the following exercises, find the critical numbers in the domains of the following functions.

$y = ln (x - 2)$

For the following exercises, find the critical numbers in the domains of the following functions.

$y = 4 - x^{2}$

For the following exercises, find the critical numbers in the domains of the following functions.

$y = \frac{x ^{2} - 1}{x ^{2} + 2 x - 3}$

For the following exercises, find the critical numbers in the domains of the following functions.

$y = x + \frac{1}{x}$