Let f(x) = −x^4 − 2x^3 + 6x + 5.

Question

a) Find the interval(s) where f(x)

 is concave up. (Enter your answer using interval notation.)

 

(b)

Find the interval(s) where f(x)

 is concave down. (Enter your answer using interval notation.)

 

(c)

Find the x-value(s) of the inflection points of f(x).

 (Enter your answers as a comma-separated list. If there are no inflection points, enter NONE.)

Answer 1

f(x) = -x⁴ - 2x³ + 6x + 5

f'(x) = -4x³ - 6x² + 6

f"(x) = -12x² - 12x = -12x(x + 1) = 0 when x = 0 or -1

When x < -1, f"(x) < 0, So, f is concave down.

When -1 < x < 0, f"(x) > 0. So, f is concave up.

When x > 0, f"(x) < 0. So, f is concave down.

Inflection points are (-1, f(-1)) and (0, f(0)).