Maximum of f(x)=(1-x)(1+x)(1+x)

Maximum of f(x) = (1-x)(1+x)(1+x)

by

Angie, Beth, and Teisha

Problem:
Find the maximum of

in the interval [0,1].

Solution:
There are several ways to find the maximum of

In this essay, we are going to find the maximum by using a graphing tool, a spreadsheet, calculus, and the arithmetic mean -- geometric mean inequality.

First, we graphed the above equation using Algebra Expresser to get an estimate for the maximum in the interval [0,1].

We kept zooming in on the graph until we were satisfied with your estimate. We found that f(x) is at a maximum when x=.33 in the interval [0,1].

Next, we developed a spreadsheet in Excel to find the maximum volume. Since we knew that the value for x is approximately .33, we began our spreadsheet with x = .33 and did iterations of .001 and .0001. Click here for the spreadsheet. We found that the maximum occurred when x = .3333.

The third way that we solved this problem is by using calculus.

Now, we need to set f'(x)=0 to get our critical values.

Next, we used the second derivative test to determine where the relative maximum occurred.

Since this is a positive value, the graph of f(x) is concave upward. Thus, this is a minimum value.

Since this is a negative value, the graph of f(x) is concave downward. Thus, f(1/3) is a relative maximum.

The last approach that we used is the Arithmetic Mean -- Geometric Mean Inequality. To find the maximum of a function using the AM--GM Inequality, the arithmetic mean must be equal to a constant and (1-x)=(1+x)=(1+x) in the geometric mean. To get the arithmetic mean to equal a constant, we did the following:

Our inequality is

We also know that in order to have a maximum

Thus, when x= 1/3, we f(x) is a maximum.

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