Question

Consider the following. (If an answer does not exist, enter DNE.)

f '(x) = x² + x − 30

(a)

Find the open intervals on which f ′(x) is increasing or decreasing. (Enter your answers using interval notation.)

increasing

(−12​,∞)

decreasing

(−∞,−12​)

(b)

Find the open intervals on which the graph of f is concave upward or concave downward. (Enter your answers using interval notation.)

concave upward

concave downward

(c)

Find the x-values of the relative extrema of f. (Enter your answers as a comma-separated list.)

relative maximum x =

relative minimum x =

Find the x-values of the inflection points of f. (Enter your answers as a comma-separated list.)

Answer 1

f '(x) = x^2 + x − 30 = (x + 6)(x - 5)

f''(x) = 2x + 1

(a) f'(x) increasing => f''(x)> 0 , interval = (-1/2 , infinity)

f'(x) decreasing => f''(x)< 0 , interval = (-infinity , -1/2)

(b) f concave upward => f''(x)> 0 , interval = (-1/2 , infinity)

f concave downward => f''(x)< 0 , interval = (-infinity , -1/2)

(c) relative extrema of f , f'(x) = 0 , x = -6 , 5

relative maximum x = -6

relative minimum x = 5

Inflection points => f''(x) = 0 ,

x = -1/2