Provide an appropriate response.
-Find the critical values and determine the intervals where f(x) is decreasing for f(x) = 3(x - 4)2/3 + 6.

Question

Quizplus · Accepted Answer

To find the critical values, we take the derivative of f(x) and set it equal to zero:
f(x) = 3(x - 4)^(2/3) + 6
f'(x) = 2(x - 4)^(-1/3)
2(x - 4)^(-1/3) = 0
x - 4 = 0
x = 4
So the only critical value is x = 4.

To determine where f(x) is decreasing, we need to look at the sign of the derivative. Since f'(x) = 2(x - 4)^(-1/3) is never negative, f(x) is never decreasing. Therefore, there are no intervals where f(x) is decreasing. To find the critical values, we need to take the derivative of f(x) and set it equal to zero.

f'(x) = 2(x - 4)^(-1/3)

Setting this equal to zero gives us:

2(x - 4)^(-1/3) = 0

This has no solution since a non-zero number raised to a negative power is never equal to zero.

Therefore, f(x) has no critical values.

To determine the intervals where f(x) is decreasing, we need to check the sign of f'(x) for values of x in different intervals.

When x < 4: f'(x) is negative since (x - 4)^(-1/3) is negative. This means that f(x) is decreasing in this interval.

When x > 4: f'(x) is positive since (x - 4)^(-1/3) is positive. This means that f(x) is increasing in this interval.

Therefore, f(x) is decreasing only in the interval x < 4.

The correct choice is B.

Provide an Appropriate Response