What are the points of inflection, if any, of $f (x) = 5 x^{3} + 30 x^{2} - 432 x$ $f(x) = 5x^3 + 30x^2 - 432x$ ?

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Answer 1 · 2018-05-22T01:35:03

$(- 2, 944)$ $(-2,944)$

points of inflection are where $f''(x)$ changes signs

in this problem, $f'(x)=15x^2+60x-432$ , $f''(x)=30x+60$

since $f''(x)$ is linear, you can find where it changes signs by setting it equal to 0.

$f''(x)=30x+60=0$
$x=-2$

if you plug in x-values near -2, you can see that $f''(x)$ changes signs at x=-2:

$f''(-1.9)=3>0$
$f''(-1.99)=0.3>0$

$f''(-2.1)=-3<0$
$f''(-2.01)=-0.3<0$

to find point of inflection: $f(-2)=944$
the point of inflection is $(-2,944)$

check with the graph of $f(x)$ :
graph{5x^3+30x^2-432x [-14, 10, -10000, 10000]}

it seems to change concavity around $x=-2$

What are the points of inflection, if any, of f(x) = 5x^3 + 30x^2 - 432x f(x)=5x3+30x2−432xf(x) = 5x^3 + 30x^2 - 432x ?