How do you determine whether the function #f(x) = 6x^3+54x-9# is concave up or concave down and its intervals?

Question

William Abdalla · Accepted Answer

You can study the sign of the second derivative:

You start with the first derivative:
So:
#f'(x)=18x^2+54#
then:
#f''(x)=36x#
set #f''(x)>0#
so:
#36x>0# when #x>0#
or visually:

Luke Alvarez · Answer

undefined To determine the concavity of the function $ f(x) = 6x^3 + 54x - 9 $, you need to find its second derivative and then analyze its sign. First, find the first derivative of $ f(x) $, then find the second derivative. Given $ f(x) = 6x^3 + 54x - 9 $, the first derivative, $ f'(x) $, is: $$ f'(x) = 18x^2 + 54 $$ Now, differentiate $ f'(x) $ to find the second derivative: $$ f''(x) = 36x $$ To determine concavity, analyze the sign of $ f''(x) $: - If $ f''(x) > 0 $ for all $ x $ in an interval, then the function is concave up in that interval. - If $ f''(x) < 0 $ for all $ x $ in an interval, then the function is concave down in that interval. Since $ f''(x) = 36x $, it changes sign at $ x = 0 $. Hence, the function changes concavity at $ x = 0 $. - $ f''(x) > 0 $ for $ x > 0 $, so the function is concave up for $ x > 0 $. - $ f''(x) < 0 $ for $ x < 0 $, so the function is concave down for $ x < 0 $. Therefore, the function $ f(x) = 6x^3 + 54x - 9 $ is concave up for $ x > 0 $ and concave down for $ x < 0 $.