Answer:
The maximum value of f(x) occurs at:

And is given by:

Step-by-step explanation:
Answer:
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:
![\displaystyle f'(x) = \frac{d}{dx} \left[ x^a\left(2-x\right)^b\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft%5B%20x%5Ea%5Cleft%282-x%5Cright%29%5Eb%5Cright%5D)
By the Product Rule:
![\displaystyle \begin{aligned} f'(x) &= \frac{d}{dx}\left[x^a\right] (2-x)^b + x^a\frac{d}{dx}\left[(2-x)^b\right]\\ \\ &=\left(ax^{a-1}\right)\left(2-x\right)^b + x^a\left(b(2-x)^{b-1}\cdot -1\right) \\ \\ &= x^a\left(2-x\right)^b \left[\frac{a}{x} - \frac{b}{2-x}\right] \end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20f%27%28x%29%20%26%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%5Bx%5Ea%5Cright%5D%20%282-x%29%5Eb%20%2B%20x%5Ea%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%5B%282-x%29%5Eb%5Cright%5D%5C%5C%20%5C%5C%20%26%3D%5Cleft%28ax%5E%7Ba-1%7D%5Cright%29%5Cleft%282-x%5Cright%29%5Eb%20%2B%20x%5Ea%5Cleft%28b%282-x%29%5E%7Bb-1%7D%5Ccdot%20-1%5Cright%29%20%5C%5C%20%5C%5C%20%26%3D%20x%5Ea%5Cleft%282-x%5Cright%29%5Eb%20%5Cleft%5B%5Cfrac%7Ba%7D%7Bx%7D%20-%20%5Cfrac%7Bb%7D%7B2-x%7D%5Cright%5D%20%5Cend%7Baligned%7D)
Set the derivative equal to zero and solve for <em>x: </em>
![\displaystyle 0= x^a\left(2-x\right)^b \left[\frac{a}{x} - \frac{b}{2-x}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%200%3D%20x%5Ea%5Cleft%282-x%5Cright%29%5Eb%20%5Cleft%5B%5Cfrac%7Ba%7D%7Bx%7D%20-%20%5Cfrac%7Bb%7D%7B2-x%7D%5Cright%5D)
By the Zero Product Property:

The solutions to the first equation are <em>x</em> = 0 and <em>x</em> = 2.
First, for the second equation, note that it is undefined when <em>x</em> = 0 and <em>x</em> = 2.
To solve for <em>x</em>, we can multiply both sides by the denominators.

Simplify:

And solve for <em>x: </em>

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 <em>a</em> and <em>b</em> > 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 <em>a</em> = 2 and <em>b</em> = 3. Then:

The critical point will be at:

Testing <em>x</em> = 0.5 and <em>x</em> = 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:
