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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:
By the Product Rule:
Set the derivative equal to zero and solve for <em>x: </em>
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: