Chapter 9.3, Problem 10QR

To determine

To find the maximum value of f and where it occurs.

Answer to Problem 10QR

The absolute maximum value at (0,2).

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\frac{x+\sin x}{x}$

Calculation :

Take derivative of the function

  $\begin{array}{l}{f}^{\prime }\left(x\right)=\frac{x(1+\cos x)-\left(x+\sin x\right)}{x^2}\\ f^{\prime }\left(x\right)=\frac{x\cos x-\sin x}{x^2}\left[\text{ simplify}\right]\end{array}$

Equate the derivative to 0.

  $\frac{x\cos x-\sin x}{x^2}=0$

Has no zeros but is undefined at $x=0$ the values of fat this one critical point is

  $\text{critical point value }f\left(0\right)=2\left[\underset{x\to 0}{\lim }\frac{\sin x}{x}=1\right]$

Since the domain of the function is all real numbers hence it has no absolute end points.

Therefore,

The absolute maximum value at (0, 2).