Question

Find any critical numbers for the function f(x)=3x 3 −3x 2+2 and then use the second derivative test to decide whethe the critical numbers lead to relative maxima or relative minima. If the second derivative test gives no information, use the first derivative test instead. Select the correct choice below and fill in any answer boxes within your choice.

A. The critical number(s) of f(x) is/are x=_____
B There are no critical numbers.
There is a relative at the______ smaller critical number.
There is a relative at the______ larger critical number.

Answer 1

The function f(x)=3x^3−3x^2+2 has two critical numbers; x = 0, which leads to a relative maximum, and x = 2/3, which leads to a relative minimum, as determined by the second derivative test.

Step-by-step explanation:

To find the critical numbers for the function f(x)=3x^3 −3x^2+2, we first calculate its first derivative, f'(x). The critical numbers are the values of x for which the first derivative is zero or undefined. After finding the critical numbers, the second derivative test can be applied to determine the nature of these points, whether they're relative maxima or minima.

First, let's find f'(x):

f'(x) = d/dx (3x^3 - 3x^2 + 2) = 9x^2 - 6x

To find critical numbers, set f'(x) = 0:

9x^2 - 6x = 0
x(9x - 6) = 0

This gives us two critical numbers: x = 0 and x = 2/3.

Next, we calculate the second derivative, f''(x):

f''(x) = d^2/dx^2 (3x^3 - 3x^2 + 2) = 18x - 6

Now, apply the second derivative test at each critical number:

• f''(0) = 18(0) - 6 = -6, which is less than 0, indicating a relative maximum.
• f''(2/3) = 18(2/3) - 6 = 6, which is greater than 0, indicating a relative minimum.

If the second derivative test were inconclusive, we would then apply the first derivative test by analyzing the sign of the first derivative around the critical numbers.