a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$ f(x)=\sqrt{25-x^{2}}, \quad-5 \leq x \leq 5 $$

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## Recommended Questions

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

g(x)=\frac{x^{2}}{4-x^{2}}, \quad-2< x \leq 1

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(x)=(x+1)^{2}, \quad-\infty<x \leq 0

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

k(x)=x^{3}+3 x^{2}+3 x+1, \quad-\infty<x \leq 0

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(x)=2 x-x^{2}, \quad-\infty<x \leq 2

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

g(x)=x^{2}-4 x+4, \quad 1 \leq x<\infty

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

h(x)=\frac{x^{3}}{3}-2 x^{2}+4 x, \quad 0 \leq x<\infty

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

g(x)=-x^{2}-6 x-9, \quad-4 \leq x<\infty

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(t)=12 t-t^{3}, \quad-3 \leq t<\infty

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(x)=\sqrt{x^{2}-2 x-3}, \quad 3 \leq x<\infty

$$

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(t)=t^{3}-3 t^{2}, \quad-\infty< t \leq 3

$$

Determine the global extreme values of the function on the given domain.

$$f(x, y)=x^{2}+2 y^{2}, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1$$

a. Find the absolute maximum and minimum values of each function

on the given interval.

b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4$$

Determine the global extreme values of the function on the given domain.

$$f(x, y)=x^{3}-2 y, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1$$

Determine the global extreme values of the function on the given set without using calculus.

$$f(x, y)=\left(x^{2}+y^{2}+1\right)^{-1}, \quad 0 \leq x \leq 3, \quad 0 \leq y \leq 5$$

a. Find the critical points of the following functions on the domain or on the given interval.

b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither.

$$f(x)=12 x^{5}-20 x^{3} \text { on } [-2,2]$$

Determine the global extreme values of the function on the given domain.

$$f(x, y)=x^{3}+x^{2} y+2 y^{2}, \quad x, y \geq 0, \quad x+y \leq 1$$

Determine the global extreme values of the function on the given domain.

$$

f(x, y)=x^{3}+y^{3}-3 x y, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1

$$

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

$$

g(x)=x^{4}-4 x^{3}+4 x^{2}

$$

a. Find the absolute maximum and minimum values of each function

on the given interval.

b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$g(x)=x e^{-x}, \quad-1 \leq x \leq 1$$

Determine the global extreme values of the function on the given domain.

$$f(x, y)=5 x-3 y, \quad y \geq x-2, \quad y \geq-x-2, \quad y \leq 3$$

Domain and Range from a Graph A function $f$ is given. (a) Sketch a graph of $f .$ (b) Use the graph to find the domain and range of $f .$

$f(x)=x-2, \quad-2 \leq x \leq 5$

? Domain and Range from a Graph A function $f$ is given. (a) Sketch a graph of $f$. (b) Use the graph to find the domain and range of $f$.

$$

f(x)=x-2, \quad-2 \leq x \leq 5

$$

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Determine the global extreme values of the function on the given domain.

$$f(x, y)=x^{2}+y^{2}-2 x-4 y, \quad x \geq 0, \quad 0 \leq y \leq 3, \quad y \geq x$$

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

$$

g(x)=x^{2} \sqrt{5-x}

$$

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

$$g(x)=\left\{\begin{array}{ll}

-x, & 0 \leq x<1 \\

x-1, & 1 \leq x \leq 2

\end{array}\right.$$

a. Find the absolute maximum and minimum values of each function

on the given interval.

b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$g(x)=e^{-x^{2}}, \quad-2 \leq x \leq 1$$

a. Find the critical points of $f$ on the given interval.

b. Determine the absolute extreme values of $f$ on the given interval.

c. Use a graphing utility to confirm your conclusions.

$$f(x)=x^{1 / 2}\left(x^{2} / 5-4\right) ;[0,4]$$

(a) graph each function. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing.

$$

F(x)=-4 x^{2}+20 x-25

$$

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)=5 x^{2}-5 x$$

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.

$$

g(x)=2+\sqrt{3 x-5}

$$