Determine the extreme values of the function f(x)=x^2-8x+9, where -1≤x≤5. What is the absolute minimum and the absolute maximum?

Answer 1

We're given the function f(x) = x² -8x + 9 over the closed interval [-1,5]

By the Extreme Value Theorem, it must have an absolute maximum and absolute minimum on the interval.

To find any local extrema, we first find the critical values by setting the first derivative to zero and solving:

f^'(x) = 2x - 8 = 0, solving for x gives a critical value on (4, -7). Since f^''2) = 2 > 0, this is a local minimum.

To find the absolute maximum and minimum, we have to evaluate the function at the endpoints:

We get f(-1) = 18, and f(5) = -6.

So among the three points (-1, 18), (4, -7) and (5, -6)

(4,-7) is the absolute min

(-1, 18) is the absolute max