Final answer:
To find the absolute maximum and minimum of f(x) = 5x^2 - 100x + 750 on the interval [4, 14], we found the derivative, identified the critical point at x = 10, and evaluated the function at x = 10, 4, and 14. The absolute maximum value is 490 at x = 14, and the absolute minimum value is 250 at x = 10.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) =5x2 − 100x + 750 on the interval [4, 14], we must consider both the critical points of the function within this interval and the values of the function at the endpoints of the interval.
Step 1: Find the derivative of the function
First, we find the derivative of the function, f'(x), to determine the critical points.
f'(x) = 10x - 100
Step 2: Solve for critical points
Next, we solve the equation f'(x) = 0 to find the critical points within the interval.
10x - 100 = 0
x = 10
Step 3: Evaluate the function at the critical point and endpoints
We then evaluate f(x) at x = 10, as well as at the endpoints of the interval, x = 4 and x = 14.
f(4) = 5(4)2 - 100(4) + 750 = 390
f(10) = 5(10)2 - 100(10) + 750 = 250
f(14) = 5(14)2 - 100(14) + 750 = 490
Absolute Maximum and Minimum
The absolute maximum value of f(x) on the interval [4, 14] is 490 at x = 14. The absolute minimum value is 250 at x = 10.