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

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

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

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.