Question

A formula for the derivative of a function f is given. How many critical numbers does \(f\) have?

49. \(f'\left( x \right) = 5{e^{ - 0.1\left| x \right|}}\sin x - 1\)

Short answer

The function \(f\) contains 10 critical numbers.

Step-by-step solution

Definition of critical number

A criticalnumber is defined as the number \(c\) in the domainof \(f\) which is either \(f'\left( c \right) = 0\) or \(f'\left( c \right)\) does not exist.

Determine the critical number of \(f\)

The procedure to draw the graph of the equation by using the graphing calculator is as follows:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(5{e^{ - 0.1\left| x \right|}}\sin x - 1\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of \(f'\left( x \right)\) as shown below:

It is observed from the graph of \(f'\left( x \right)\) that it contains 10 zeros and exists everywhere.

Thus, the function \(f\) contains 10 critical numbers.