Consider the function below.

Question

f(x) = x√x + 6

(a) What is the domain of the function? (Enter your answer using interval notation.)

(b) Find the interval(s) where the function is increasing. (Enter your answer using interval notation.)

Find the interval(s) where the function is decreasing. (Enter your answer using interval notation.)

 

(c) Find the x-coordinate(s) of any local minima. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Local minima at x = 

Find the x-coordinate(s) of any local maxima. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Local maxima at x = 

Answer 1

f(x) = x√(x+6)

Domain: x+6 ≥ 0 So, x ≥ -6.

f'(x) = √(x+6) + x(1/2)(x+6)^-1/2 = √(x+6) + x / (2√(x+6)) = [2(x+6) + x] / (2√(x+6)) = [3x+12] / (2√(x+6))

f'(x) = 0 when x = -4

When -6 < x < -4, f'(x) < 0. So, f is decreasing.

When x > -4, f'(x) > 0. So, f is increasing.

Rel min when x = -4

No rel max