If a and b are positive numbers, find the maximum value of f(x) = x^a(2 − x)^b on the interval 0 ≤ x ≤ 2.

The maximum value of f(x) occurs at:

And is given by:

Step-by-step explanation:

Step-by-step explanation:

We are given the function:

And we want to find the maximum value of f(x) on the interval [0, 2].

First, let's evaluate the endpoints of the interval:

And:

Recall that extrema occurs at a function's critical points. The critical points of a function at the points where its derivative is either zero or undefined. Thus, find the derivative of the function:

By the Product Rule:

Set the derivative equal to zero and solve for x:

By the Zero Product Property:

The solutions to the first equation are x = 0 and x = 2.

First, for the second equation, note that it is undefined when x = 0 and x = 2.

To solve for x, we can multiply both sides by the denominators.

Simplify:

And solve for x:

So, our critical points are:

We already know that f(0) = f(2) = 0.

For the third point, we can see that:

This can be simplified to:

Since a and b > 0, both factors must be positive. Thus, f(2a / (a + b)) > 0. So, this must be the maximum value.

To confirm that this is indeed a maximum, we can select values to test. Let a = 2 and b = 3. Then:

The critical point will be at:

Testing x = 0.5 and x = 1 yields that:

Since the derivative is positive and then negative, we can conclude that the point is indeed a maximum.

Therefore, the maximum value of f(x) occurs at:

And is given by:

what are the statements

step-by-step explanation: