Calculus question

Question

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

f(x)= (x^2-64)/(x^2+64). [-64, 64]

Answer 1

The absolute maximum and absolute minimum of a function, provided that it is continuous on the closed interval, will occur either at one of its critical values(where the function's derivative is equal to zero or where the function's derivative is undefined) or at one of its endpoints. This is known as the Extreme Value Theorem.

To find the absolute maximum and absolute minimum for the given function, first verify that it is continuous on the closed interval. Then take the derivative and find the critical values. Finally, make an ordered list(smallest x value to largest) evaluating the original function at each of its end points and critical values. The largest value will be the absolute maximum, the smallest will be the absolute minimum:

1. f(x) is a polynomial which is continuous for all real numbers, and will therefore be continuous on the closed interval [ -64, 64].
2. f(x) = x⁴ - 128x² - 4,096(multiplied out to make it easier to differentiate), so the derivative is:

f'(x) = 4x³ - 256x.

Find critical values: 4x³ - 256x = 0, x(4x² - 256) = 0, then x = 0 or 4x² - 256 = 0,

So x = 0, x = -8, and x = 8 are the critical values.

Evaluate f(x) at each endpoint and at each critical value within the closed interval. If and critical numbers are outside the given closed interval, you do not need to evaluate the function at those values:

f(-64) = 16,773,120

f(-8) = 0

f(0) = -4,096

f(8) = 0

f(64) = 16,773,120

In this case, the absolute maximum is 16,773,120 and it occurs at two places, when x = -64 and when x = 64 (the left and right endpoints of the closed interval), and the absolute minimum is -4,096 when x = 0.

You can graph the function and its derivative on a graphing calculator to verify.

I hope this helps. Please let me know if you have any further questions.

Answer 2

There's a procedure for finding the absolute min and max of a continuous function on a closed interval. Maybe you've seen it before.

Find the critical points of the function within the interval. Then evaluate the function at each of these critical points and at the endpoints of the interval. The smallest of these values is the min, the largest is the max.