Optimization > Maximize a function

## How to Maximize a Function: General Steps

General steps to maximize a function on a closed interval [a, b]:

- Find the first derivative,
- Set the derivative equal to zero and solve,
- Identify any values from Step 2 that are in [a, b],
- Add the endpoints of the interval to the list,
- Evaluate your answers from Step 4: The largest function value is the maximum.

**Example problem #1:** Find the maximum of the function f(x) = x^{4} – 8x^{2} + 3 on the interval [-1, 3].

Step 1: **Find the first derivative. ** Our function is f(x) = x^{4} – 8x^{2} + 3, so we can take the derivative with the power rule, giving:

f′(x) = 4x^{3} – 16x.

Step 2: **Set the derivative equal to zero and solve,** using algebra.

- 4x
^{3}– 16x = 0. - x
^{3}– 4x = 0 (Dividing by 4). - x(x – 2)(x + 2) = 0((Factoring)

Solving, the critical points are -2, 0 and 2.

If algebra isn’t your strong point, use Symbolab’s algebra calculator to solve for 0. Here’s the format:

Step 3: **Identify any values from Step 2 that are in [a, b].** These values are all potential maximums and are called critical points.

The interval given in the question was [-1, 3]. Of the three critical points we found: -2, 0 and 2, only two are in the specified interval: **0 and 2.**

Step 4: **Add the endpoints of the interval to the list.**

The endpoints of the interval are given in the question as -1 and 3, which gives a total of four possibilities: -1, 0, 2, and 3.

Step 5: **Plug the values from Step 4 into the original function **(if you have a TI-89, creating a table of values makes this step a lot faster!):

- f(-1) = -1
^{4}– 8(-1)^{2}+ 3 = -4, - f(0) = 0
^{4}– 8(0)^{2}+ 3 = 3, - f(2) = 2
^{4}– 8(2)^{2}+ 3 = -13, - f(3) = 3
^{4}– 8(3)^{2}+ 2 = 12.

The largest function value is the maximum; The function has a maximum at x = 3 on the interval [-1, 3].

We can confirm this with a graph (I used Desmos.com):

## Why is it Useful?

Finding the largest value for a function has dozens of practical uses in real life, especially in business, engineering and the sciences. For example, businesses always want to how to maximize a function, specifically to find a maximum profit: that sweet spot where a set of inputs (like workers and capital) can be maximized to produce the most profit.

For some practical examples of how to maximize a function, visit the Optimization page.