On what interval is #f(x)=6x^3+54x-9# concave up and down?

Question

Paisley Johnson · Accepted Answer

A function is concave up when the second derivative is positive, it is concave down when it is negative, and there could be an inflection point when it is zero.

#y'=18x^2+54#

#y''=36x+54#

so:

#y''>0rArrx> -54/36rArrx> -3/2#.

In #(-3/2,+oo)# the concave is up,

in #(-oo,-3/2)#the concave is down,

in #x=-3/2# there is an inflection point.

Emily Ammerman · Answer

undefined To determine the intervals where $ f(x) = 6x^3 + 54x - 9 $ is concave up and concave down, we need to find the second derivative of the function and analyze its sign. First, find the first derivative of $ f(x) $ with respect to $ x $: $$ f'(x) = 18x^2 + 54 $$ Next, find the second derivative of $ f(x) $: $$ f''(x) = 36x $$ To determine concavity, analyze the sign of $ f''(x) $: - If $ f''(x) > 0 $, the function is concave up. - If $ f''(x) < 0 $, the function is concave down. Since $ f''(x) = 36x $, it's positive when $ x > 0 $ (concave up) and negative when $ x < 0 $ (concave down). Therefore, $ f(x) = 6x^3 + 54x - 9 $ is concave up for $ x > 0 $ and concave down for $ x < 0 $.