Question

Find the absolute maximum and absolute minimum values of f on each interval.

f(x) =−5x2 + 30x + 550
(a) ( 3, 8 )
Absolute maximum:
Absolute minimum:

Answer 1

To find the absolute maximum and absolute minimum values of the function f(x) = -5x^2 + 30x + 550 on the interval (3, 8), we find the critical point and evaluate the function at the endpoints of the interval.

Step-by-step explanation:

To find the absolute maximum and absolute minimum values of the function f(x) = -5x^2 + 30x + 550 on the interval (3, 8), we need to first find the critical points of the function. The critical points are the values of x where the derivative of the function is either zero or undefined. Taking the derivative of f(x), we get f'(x) = -10x + 30. Setting this equal to zero and solving for x, we find x = 3. Therefore, the only critical point on the interval (3, 8) is x = 3.

Next, we need to evaluate the function f(x) at the critical point and the endpoints of the interval. Substituting x = 3 into the function, we get f(3) = -5(3)^2 + 30(3) + 550 = 725. Evaluating f(8), we get f(8) = -5(8)^2 + 30(8) + 550 = 370. Finally, evaluating f(3) and f(8) gives us the absolute maximum and absolute minimum values on the interval (3, 8):
Absolute maximum: 725
Absolute minimum: 370