Unanswered boxes are questions. Consider the function below. C(x) = x1/7(x + 8)

Question

Unanswered boxes are questions.

Consider the function below. C(x) = x1/7(x + 8) (a) 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

C(x) = x^1/7 (x + 8)

differentiate wrt x

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

(a)

For increasing function, C'(x) > 0

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

7x + x + 8 > 0

8x > -8

x > -1

x = (-1, infinity)

For decreasing function, C'(x) < 0

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

7x + x + 8 < 0

8x < -8

x < -1

x = (-infinity, -1)

(b)

local minimum & maximum values exist at critical points

for critical points, C'(x) = 0

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

7x + x + 8 = 0

8x = -8

x = -1

C''(x) = 1/7x^-6/7 + 1/7x^-6/7 - 6/49 x^-13/7 (x + 8)

C''(-1) = 1/7 + 1/7 + 6/49*7 = 8/7 > 0

hence we have local minimum at x = -1

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