Consider the function f(x)=x^4-18x^2+5, -2≤x≤7

Question

The function has an absolute minimum value equal to -76 and an absolute maximum value equal to ____

I am having trouble finding the maximum value

Answer 1

Critical numbers are where f'(x) = 0, where f'(x) DNE, and at the endpoints of the domain intervals.

Take the derivative by using the power rule:

f'(x) = 4x³ -36x

Critical Numbers where f'(x) DNE

There are none - f'(x) is continuous for all x

Critical Numbers where f'(x) = 0

4x³ -36x = 0

4x(x² - 9) = 0

4x(x +3)(x - 3) = 0

x = 0, -3, 3 but x = -3 is not in the domain so x = 0, 3

Critical Numbers at the endpoints

x = -2

x = 7

To find global extrema values, plug in the critical numbers into the original function:

for x = -2, f(-2) = -51

for x = 0, f(0) = 5

for x = 3, f(3) = -76

for x = 7, f(7) = 1524

The global (absolute) minimum is the point (3, -76) where the absolute minimum value is -76

The global (absolute) maximum is the point (7, 1524) where the absolute maximum value is 1524