Question

Determine where the absolute extrema of f(x)=−5x²+ 11x on the interval [1,3] occur. 1. The absolute maximum occurs at x = ? 2. The absolute minimum occurs at x =?

Answer 1

The absolute maximum of the function f(x) = -5x² + 11x on the interval [1,3] occurs at x = 1, and the absolute minimum occurs at x = 3.

Step-by-step explanation:

To determine where the absolute extrema of the function f(x) = -5x² + 11x on the interval [1,3] occur, we first need to find the critical points of the function in the interval and then evaluate the function at those points as well as at the endpoints of the interval.

First, let's find the derivative of the function:

f'(x) = d/dx (-5x² + 11x) = -10x + 11.

Now, set the derivative equal to zero to find the critical points:

-10x + 11 = 0
x = 1.1

However, since 1.1 is not in the interval [1,3], we do not consider it. We then evaluate the function at the endpoints of the interval:

f(1) = -5(1)² + 11(1) = 6,
f(3) = -5(3)² + 11(3) = -24.

Comparing the values at x = 1 and x = 3, we find that the absolute maximum occurs at x = 1 with f(1) = 6 and the absolute minimum occurs at x = 3 with f(3) = -24.