## Related questions with answers

**For each function,**

**(a) find $y^{\prime}=f^{\prime}(x)$.**

**(b) find the critical values.**

**( c ) find the critical points.**

**(d) find intervals of $x$-values where the function is increasing and where it is decreasing.**

**(e) classify the critical points as relative maxima, relative minima, or horizontal points of inflection. In each case, check your conclusions with a graphing calculator.**

$y=\frac{x^4}{4}-\frac{x^3}{3}-2$

Solution

Verifieda. Let the function $y$ be: $\frac{x^4}{4}-\frac{x^3}{3}-2$

Taking the derivative of $y$, we have

$\begin{aligned} y'&=\frac{d}{dx}\left(\frac{x^4}{4}+\frac{x^3}{3}-2\right)&&\text{Derivative form}\\&=\frac{d}{dx}\left(\frac{x^4}{4}\right)-\frac{d}{dx}\left(\frac{x^3}{3}\right)-\frac{d}{dx}(2)&&\text{Sum rule}\\&=\frac{4x^3}{4}+\frac{3x^2}{3}-0&&\text{Apply common derivative}\\&=x^3-x^2&&\text{Simplify} \end{aligned}$

Therefore,

$\boxed{\bold{y'=x^3-x^2}}$

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