Question

Find the absolute minimum and maximum values of f(x)=3x²−9/x.

a) No absolute minimum or maximum
b) Absolute minimum at x=−3, absolute maximum at x=3
c) Absolute minimum at x=3, absolute maximum at x=−3
d) Absolute minimum at x=−3 and x=3

Answer 1

To find the absolute minimum and maximum values of the function f(x) = 3x² - 9/x, we need to find the critical points and analyze the endpoints of the interval. The correct answer is option C.

Step-by-step explanation:

To find the absolute minimum and maximum values of the function f(x) = 3x² - 9/x, we need to find the critical points and analyze the endpoints of the interval. Firstly, we find the derivative of f(x):

f'(x) = 6x + 9/x²

To find the critical points, we set f'(x) equal to 0 and solve for x:

6x + 9/x² = 0

Next, we solve this equation to find the x-values of the critical points. By analyzing the behavior of the function on the given interval and checking the signs of the derivative, we can determine if these critical points correspond to relative minimum or maximum values.

Finally, we evaluate the function at the critical points and endpoints to find the absolute minimum and maximum values of f(x).