# Perform the first and second derivative test on the following function and find all local maxes...

## Question:

Perform the first and second derivative test on the following function and find all local maxes and mins, inflection points, domains of increasing and decreasing, and concavity.

{eq}f(x) = 3x^{\frac{1}{3}} - \frac{3}{4}x^{\frac{4}{3}} {/eq}

## Critical Points:

With the first derivative, we can know the critical points that a function has, we can also classify them into Maximum and Minimum, instead, with the second derivative we know the inflection points.

We have the function

{eq}f(x) = 3x^{\frac{1}{3}} - \frac{3}{4}x^{\frac{4}{3}} \\ {/eq}

Domain:

{eq}[0, \infty) \\ {/eq}

Differentiating the function

{eq}f'(x)=-{\frac {x-1}{{x}^{2/3}}} \\ {/eq}

{eq}f'(x)=0 {/eq} when {eq}x=1 {/eq} and {eq}f'(x)= DNE {/eq} when {eq}x=0 {/eq}

Critical points:

{eq}(0,0) \\ (1, \frac{9}{4} ) \\ {/eq}

First derivatives test:

{eq}\begin{array}{r|D{.}{,}{5}} Interval & {0<x<1} & {1<x<\infty } \\ \hline Test \space{} value & \ x=0.5 & \ x=2 \\ Sign \space{} of \ f'(x) & \ f'(0.5)>0 & \ f'(2)<0 \\ Conclusion & increasing & decreasing \\ \end{array} \\ {/eq}

Local maximum:

{eq}(1, \frac{9}{4} ) \\ {/eq}

Local minimum:

{eq}(0,0) \\ {/eq}

Differentiating the function

{eq}f''(x)=- \frac{1}{3}\,{\frac {x+2}{{x}^{5/3}}} \\ {/eq}

{eq}f''(x)=0 {/eq}

{eq}x=-2 {/eq} It is not within the domain of the function, consequently, the function has no inflection points.

{eq}\begin{array}{r|D{.}{,}{5}} Interval & {0<x<\infty } \\ \hline Test \space{} value & \ x=1 \\ Sign \space{} of \ f'' (x) & \ f'' (1)<0 \\ Conclusion & concave \space down \\ \end{array} \\ {/eq} 