Chapter 1.2, Problem 87E

Solution

(a)

To find: the function is continuous on the given interval and the maximum and minimum values of the function.

The $f\left(x\right)$ is continuous on $\left[-2,4\right]$ .

The maximum value is $1$ which occurs at $x=1$ .

The minimum value is $-3$ which occurs at $x=0$ .

Given information:

The function $f\left(x\right)=x^2-3,\left[-2,4\right]$ .

Calculation:

The graph of the given function on the interval is displayed below.

Ensure that the right end of the graph contacts the right side of the screen and that the left end touches the left.

  

Confirmed, $f\left(x\right)$ is continuous on $\left[-2,4\right]$ .

Determine the graph's maximum and minimum points and their values using the maximum and minimum features:

  

The maximum value is 1 which occurs at $x=1$ .

The minimum value is $-3$ which occurs at $x=0$ .

Using a graphing calculator, $f\left(x\right)$ is continuous on $\left[-2,4\right]$ .

The maximum value is $1$ which occurs at $x=1$ .

The minimum value is $-3$ which occurs at $x=0$ .

(b)

To find: the function is continuous on the given interval and the maximum and minimum values of the function.

The function $f\left(x\right)$ is continuous on $\left[1,5\right]$ .

The maximum value is 1 which occurs at $x=1$ .

The minimum value is $0.2$ which occurs at $x=5$ .

Given information:

The function $f\left(x\right)=\frac{1}{x},\left[1,5\right]$ .

Calculation:

The graph of the given function on the interval is displayed below.

Ensure that the right end of the graph contacts the right side of the screen and that the left end touches the left.

  

Confirmed, $f\left(x\right)$ is continuous on $\left[1,5\right]$ .

Determine the graph's maximum and minimum points and their values using the maximum and minimum features:

  

The maximum value is 1 which occurs at $x=1$ . The minimum value is $0.2$ which occurs at $x=5$

Using a graphing calculator, $f\left(x\right)$ is continuous on $\left[1,5\right]$ . The maximum value is 1 which occurs at $x=1$ . The minimum value is $0.2$ which occurs at $x=5$ .

(c)

To find: the function is continuous on the given interval and the maximum and minimum values of the function.

The function $f\left(x\right)$ is continuous on $\left[-4,1\right]$ .

The maximum value is 5 which occurs at $x=-4$ .

The minimum value is 2 which occurs at $x=-1$ .

Given information:

The function $\text{f}\left(\text{x}\right)\text{=|x+1|+2,}\left[\text{-4,1}\right]$ .

Calculation:

The graph of the given function on the interval is displayed below.

Ensure that the right end of the graph contacts the right side of the screen and that the left end touches the left.

  

Confirmed, $f\left(x\right)$ is continuous on $\left[-4,1\right]$ .

Determine the graph's maximum and minimum points and their values using the maximum and minimum features:

  

The maximum value is 1 which occurs at $x=1$ . The minimum value is $0.2$ which occurs at $x=5$ Using a graphing calculator, $f\left(x\right)$ is continuous on $\left[-4,1\right]$ . The maximum value is 5 which occurs at $x=-4$ . The minimum value is 2 which occurs at $x=-1$ .

(d)

To find: the function is continuous on the given interval and the maximum and minimum values of the function.

The function $f\left(x\right)$ is continuous on $\left[-4,4\right]$ .

The maximum value is $1$ which occurs at $x=-4$ and $x=4$ .

The minimum value is 3 which occurs at $x=0$ .

Given information:

The function $f\left(x\right)=\sqrt{x^2+9},\left[-4,4\right]$ .

Calculation:

The graph of the given function on the interval is displayed below.

Ensure that the right end of the graph contacts the right side of the screen and that the left end touches the left.

  

Confirmed, $f(x)$ is continuous on $\left[-4,4\right]$ .

Determine the graph's maximum and minimum points and their values using the maximum and minimum features:

  

The maximum value is $1$ which occurs at $x=-4$ and $x=4$ . The minimum value is $3$ which occurs at $x=0$ .

Using a graphing calculator, $f\left(x\right)$ is continuous on $\left[-4,4\right]$ . The maximum value is $1$ which occurs at $x=-4$ and $x=4$ .

The minimum value is 3 which occurs at $x=0$ .