Question

second derivative help: I need to find concavity for f(x)=ln(x^4+27). I calculated the first derivative to be (4x^3)/(x^4+27). My second derivative is a cluster: (-4x^6+324x^2)/(x^4+27)^2. What's my concavity?

Answer 1

well to test concavity, look at sign of 2nd derivative

f''(x) > 0 --> concave up
f''(x) < 0 --> concave down

Answer 2

set f''(x) = 0

--> -4x^6 +324x^2 = 0

--> -4x^2 (x^4 - 81) = 0

--> -4x^2 (x^2 +9)(x^2 -9) = 0

--> -4x^2 (x^2 +9)(x+3)(x-3) = 0

real solutions: x = -3,0,3

if x<-3:
say x = -4 , then sign is negative --> concave down

if -3 < x <0 :
say x = -1, then sign is positive --> concave up

if 0 < x < 3 :
say x = 1, then sign is positive --> concave up

if x > 3:
say x=4 , then sign is negative --> concave down