Find the local and global extrema to the nearest tenth for the graph of ƒ(x) = –x3 + 9x – 1

General Idea:

In order to find the local extrema of a function f(x), we need to do the below steps:

(i) Find the first derivative of the given function

(ii) Set the first derivative to zero and solve for x to identify the critical numbers.

(iii) Draw a number line plotting the critical numbers in it, then pick a test point from each of the intervals to check whether the function is increasing or decreasing.

(iv) If , then function f(x) Decreasing in that interval. If , then the function f(x) Increasing in that interval. Based on this information we can identify the local extrema's.

Applying the concept:

Given function

Step 1: Finding the derivative of the function:

Step 2: Set the and solve for x to get the critical numbers.

Step 3: We need write the intervals based on the critical numbers

Let us pick a Test point from the interval as -2

The function will be decreasing in the interval

Let us pick a Test point from the interval as 0

The function will be increasing in the interval

Let us pick a Test point from the interval as 2

The function will be decreasing in the interval.

Conclusion:

At , function has a local minimum

At , function has a local maximum

the answer is y < -3 or   y > 3

-3y > 7 + 2

-3 y > 9

y < -3