# What are the values and types of the critical points, if any, of f(x)=2x^5(x-3)^4?

Feb 11, 2016

At $x = 0$, there is a saddle point (neither min nor max). At $x = 3$ there is a local minimum (which is $0$). At $x = \frac{5}{3}$, there is a local maximum (which is $f \left(\frac{5}{3}\right)$).

#### Explanation:

$f \left(x\right) = 2 {x}^{5} {\left(x - 3\right)}^{4}$

$f ' \left(x\right) = 10 {x}^{4} {\left(x - 3\right)}^{4} + 2 {x}^{5} \cdot 4 {\left(x - 3\right)}^{3} \left(1\right)$

$= 10 {x}^{4} {\left(x - 3\right)}^{4} + 8 {x}^{5} {\left(x - 3\right)}^{3}$

$= 2 {x}^{4} {\left(x - 3\right)}^{3} \left[5 \left(x - 3\right) + 4 x\right]$

$= 2 {x}^{4} {\left(x - 3\right)}^{3} \left(9 x - 15\right)$

$f ' \left(x\right)$ is never undefined and it is $0$ at $0$, $3$, and $\frac{15}{9} = \frac{5}{3}$.

All of these are in the domain of $f$, so they are critical points for $f$.

Sign of $f ' \left(x\right)$

{: (bb"Interval:",(-oo,0),(0,5/3),(5/3,3),(3,oo)), (darrbb"Factors"darr,"=======","======","=====","====="), (2x^4," +",bb" +",bb" +",bb" +"), ((x-3)^3, bb" -",bb" -",bb" -",bb" +"), (9x-15,bb" -",bb" -",bb" +",bb" +"), ("==========","========","======","=====","======"), (bb"Product"=f'(x),bb" +",bb" +",bb" -",bb" +") :}

The sign of $f '$ does not change at $0$, so $f \left(0\right)$ is neither minimum nor maximum.

$f '$ changes from + to - at $\frac{5}{3}$, so $f \left(\frac{5}{3}\right)$ is a local maximum.

$f '$ changes from - to + at $3$, so $f \left(3\right)$ is a local minimum.