Consider the following. f\'(x) = 7x^2 - 5 Find the intervals on which f\'(x) is
ID: 2877395 • Letter: C
Question
Consider the following. f'(x) = 7x^2 - 5 Find the intervals on which f'(x) is increasing or decreasing and the graph of f is concave upward or concave downward. Find the x-values of the relative extrema and inflections points of f. (If an answer does not exist, enter DNE.) Find the intervals on which f'(x) is increasing or decreasing. (Enter your answers using interval notation.) Find the intervals on which the graph of f is concave upward or concave downward. (Enter your answers using interval notation.) Find the x-values of the relative extrema of f. (Enter your answers as a comma-separated list.) Find the x-values of the inflection points of f. (Enter your answers as a comma-separated list.)Explanation / Answer
a)
to find f'(x) is increasing, we need to calculate f''(x)
f''(x) = 14x
f''(x) is negative if x is negative. So f'(x) is decreasing for x<0
f''(x) is positive if x is positive. So f'(x) is increasing for x>0
b)
f''(x) = 14x
if f''(x) is positive, it will be concave up. SO concave up for x>0
if f''(x) is negative, it will be concave down. SO concave down for x<0
c)
for relative extrea, put f'(x) = 0
f'(x)=7x^2-5 = 0
x = sqrt(5/7) and -sqrt(5/7)
at , x = sqrt(5/7), f''(x) is positive, so x = sqrt(5/7) is relative minimum
at , x = -sqrt(5/7), f''(x) is negative, so x = -sqrt(5/7) is relative maximum