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
<u>Step 1:</u> Finding the derivative of the function:
<u>Step 2:</u> Set the and solve for x to get the critical numbers.
<u>Step 3:</u> 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