please how to solve this?

Question

find the minimum value of the function: y = 3x + 3x² - x³ in the intervals [-2, 2]

Answer 1

Hi Learner,

There are shortcuts for this particular problem, but I'll show you the general way to attack this kind of problem.

This is an example of an absolute maximum/minimum problem. Here, however, you are specifically asked for the minimum.

You can recognize these problem types from the fact that you are asked to find the maximum or minimum or both AND you are given the interval on which this must be done.

Once you know that is the problem type, follow three steps:

1. Find the critical points of the function and evaluate the function at these points (if they exist).
2. Evaluate the function at the endpoints of the given interval.
3. Compare the evaluations from the two previous steps: the largest value is your (absolute) maximum and the smallest value is your (absolute) minimum.

Here you have,

f(x) = 3x + 3x² - x³ on the interval [-2,2]

Step 1: Find the critical points and evaluate:

For critical points, find where f'(x) = 0 or f'(x) is undefined.

Now, f'(x) = 3 + 6x - 3x²

For critical points, set f'(x) = 0 or undefined (clearly it is never undefined), thus we have

3 + 6x - 3x² = 0

=> -3(x² - 2x - 1) = 0

This does not factor, so using the quadratic formula, we get

=> x = 1 + √2 or x = 1 - √2.

Now only the second x-value is in the given interval, so we reject the first and evaluate the second.

=> f(1 - √2) = 5 - 4√2 ~= -0.657

Step 2: Evaluate f(x) at the endpoints.

f(2) = 10

and, f(-2) = 14

Step 3: Compare.

After seeing all these evaluations, f(1 - √2) = 5 - 4√2 is clearly the smallest one. This is the

minimum value of the function on the given interval.