help? derivatives-functions

Question

Answer 1

Step 1:

Graph the derivative of the function.

First derivative of the function :

1). If the graph of the function is increasing (goes up) then .

represents that the graph of the function is above -axis.

2). If the graph of the function is decreasing (goes down) then

represents that the graph of the function is below -axis.

3). at the local extremum points.

Step 2:

Observe the graph of the function .

The function is increasing in the interval , hence the derivative of the function is greater than 0.

The function is decreasing in the interval , hence the derivative of the function is less than 0.

The function is increasing in the interval , hence the derivative of the function is greater than 0.

Local maximum : .

Local minimum : .

So the derivative of the function is zero at and .

Step 3:

Graph :

Graph the function such that it satisfies the below conditions:

The derivative of the function is greater than zero for and .

The derivative of the function is less than zero for .

Solution :

Graph of the derivative of the function is