First and Second Derivative Test Question 2

What is the $x$-coordinate of the point of inflection on the graph of $y=\dfrac{1}{3}x^3+5x^2+24$ ?

$5$

$-\dfrac{10}{3}$

$-5$

$-10$

Approach

To find the points of inflection, we must find when $y''=0$.

$$ y=\frac{1}{3}x^3+5x^2+24$$ $$ y' = x^2+10x $$ $$ y'' = 2x+10 $$ $$ 0 = 2x+10 $$ $$ 0=2(x+5) $$ $$ 0 = x+5 $$ $$ x= \boxed{-5} $$

Sidenote

Typically, we would test points left and right of $x=-5$. You can do this in your head, or realize that because the multiplicity of the $(x+5)$ factor is odd, the graph passes through $x=-5$ and therefore changes sign.