Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 189: 44

$f(x)$ increasing on $\left( -\infty,-3\right )\cup (3,\infty)$ and decreasing on $(-3,-\sqrt{3})\cup ( 0,\sqrt{3})\cup (\sqrt{3}, 3)$ $f(x)$ has a local maximum at $x=-3$ and a local minimum at $x=3$.

Work Step by Step

Given $$y=\frac{x^{3}}{x^{2}-3}$$ Since \begin{align*} f'(x) &=\frac{\frac{d}{dx}\left(x^3\right)\left(x^2-3\right)-\frac{d}{dx}\left(x^2-3\right)x^3}{\left(x^2-3\right)^2}\\ &=\frac{x^4-9x^2}{\left(x^2-3\right)^2} \end{align*} Then $f'(x)=0$ for $x^4-9x^2= 0$. Hence, $$x=0,\ \ \ \ x=-3,\ \ \ \ x=3 ,\ \ \ \ x=\pm \sqrt{3}$$ Choose $x=-1,\ \ x=1$: \begin{align*} f'(-4)&> 0\\ f'(-2)&<0\\ f'(-1) &<0\\ f'(2)&<0\\ f'(4)&> \end{align*} Then $f(x)$ is increasing on $\left( -\infty,-3\right )\cup (3,\infty)$ and decreasing on $(-3,-\sqrt{3})\cup ( 0,\sqrt{3})\cup (\sqrt{3}, 3)$. Hence, $f(x)$ has a local maximum at $x=-3$ and a local minimum at $x=3$.

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