Consider the function below. C(x) = x^1/7(x + 8) Find the interval of increase.

Question

Consider the function below. C(x) = x^1/7(x + 8) Find the interval of increase. (Enter your answer using interval notation.) Find the interval of decrease. (Enter your answer using interval notation.) Find the local minimum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Find the local maximum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Find the inflection points. (x,y) = (smaller x-value) (x,y) = (larger x-value) Find the intervals where the graph is concave upward. (Enter your answer using interval notation.) Find the interval where the graph is concave downward. (Enter your answer using interval notation.)

Explanation / Answer

Solution:

Differentiate to determine the intervals of increase and decrease and relative extrema by product rule.
Differentiate the first derivative to determine the inflection points and the concavity by product rule (again!).

f'(x) = x^(1/7) + (x + 8)(1/7)(x^(-6/7))

Let f'(x) = 0 and solve for x.

0 = x^(1/7) + (x + 8)(1/7)(x^(-6/7))

0 = x + (x + 8)(1/7)

0 = x + (1/7)x + (8/7)

-8/7 = (8/7)x

x = -1

Use the sign analysis to determine the increasing and decreasing intervals. You should get:

Increasing (where f''(x) > 0) at: (-, -1) and (0, )
Decreasing (where f''(x) < 0) at: (-1, 0)

Since there is a sign change from positive to negative, the point at x = -1 is considered a max.

Second Derivative:

f''(x) = 2/(7x^(6/7)) - (6(x + 8))/(49x^(13/7))

Let f''(x) = 0 and solve for x.

0 = 2/(7x^(6/7)) - (6(x + 8))/(49x^(13/7))

Therefore, you have:

x = 6

Draw the quick second derivative sign analysis. You should get:

Concave Up (f''(x) is positive): (6, )
Concave Down (f''(x) is negative): (-, 6)
Inflection Point at x = 6.

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