How do you determine whether the function $f(x) = 2x^3 + 3x^2 - 432x$ is concave up or concave down and its intervals?

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Answer 1 · 2015-09-24T14:12:47

f(x) is concave down on $(-oo;-1/2)$ and concave up on $(-1/2;oo)$

You have to investigate the sign of the second derivative around its zero points (inflection points).

In this case,

$f'(x)=6x^2+6x-432$
$f''(x)=12x+6$

$therefore f''(x)=0iffx=-1/2$

$therefore AAx<-1/2=>f''(x)<0$ and hence the concavity is down.

$therefore AAx >-1/2=>f''(x)>0$ and hence the concavity is up.

How do you determine whether the function f(x) = 2x^3 + 3x^2 - 432x f(x) = 2x^3 + 3x^2 - 432x is concave up or concave down and its intervals?