Chapter 5, Problem 12T

(a)

To determine

To Sketch: Plot the graph of function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$ in appropriate viewing rectangle.

Explanation of Solution

The given function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$.

Substitute 0 for $x$ in above equation,

$\begin{array}{c}f\left(0\right)=\frac{\cos \left(0\right)}{1+{\left(0\right)}^2}\\ =1\end{array}$

Substitute the different values of $x$ for the additional value of function $f\left(x\right)$ as shown in table below.

x	f(x)
$\pm 1.571$	0
0	1
$\pm 2.544$	$-0.111$

Draw the graph of function as given below,

Figure (1)

Hence, Figure (1) shows the graph of the function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$ in appropriate viewing rectangle.

(b)

To determine

The function $f\left(x\right)$ is odd, even or neither through the graph.

Explanation of Solution

The given function is $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$.

Figure (1) in part (a) has symmetry about y-axis.

Hence, the function $f\left(x\right)$ is even function.

(c)

To determine

The minimum and maximum value of function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$.

Answer to Problem 12T

The function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$ has minimum value $-2.544$ and maximum value 1.

Explanation of Solution

Given:

The given function is $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$.

Calculation:

Figure (1) shows that the function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$ attains its minimum value $-2.544$ when $x$ is $-0.111$ and the function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$ attains its maximum value $1$ when $x$ is 0.

Hence, the function $f\left(x\right)=\frac{\cos \left(x\right)}{1+x^2}$ has minimum value $-2.544$ and maximum value 1.