# How to Analyze Power Functions

Power functions are a special case of polynomial functions. Power functions only have one term—a polynomial in the form $a{x}^{n}$.

Power functions are analyzed in the same way as normal polynomial functions. Here is the method:

Rule

### AnalyzingPowerFunctions

1.
Find the zeros.
2.
Find the stationary points.
3.
Find the inflection points.

Example 1

Analyze the function $f\left(x\right)=5{x}^{4}$

1.
First, you find the zeros by setting $f\left(x\right)=0$. You then get: $\begin{array}{llll}\hfill 5{x}^{4}& =0\phantom{\rule{1em}{0ex}}|÷5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {x}^{4}& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \sqrt[4]{{x}^{4}}& =\sqrt[4]{0}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

So you have a zero at $\left(0,0\right)$.

2.
You then find the maxima and minima by setting ${f}^{\prime }\left(x\right)=0$. First, you differentiate the function:
 ${f}^{\prime }\left(x\right)=20{x}^{3}$

You then set this equal to 0 to find the maxima and minima: $\begin{array}{llll}\hfill 20{x}^{3}& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

To determine if it is a maximum or a minimum, you can select two values, one to the left of $x=0$ and one to the right of $x=0$. Input these into the differentiated function ${f}^{\prime }\left(x\right)$ and interpret the sign. Choose numbers that are easy to work with, like $x=-1$ and $x=1$:

As the graph decreases first and then increases, this is a minimum.

You now need to find the $y$-value of the point, by inserting the $x$-value into the main function $f\left(x\right)$:

 $f\left(0\right)=5{\left(0\right)}^{4}=0$

Thus, you have a minimum at $\left(0,0\right)$.

3.
To find the inflection points of the function, you set ${f}^{″}\left(x\right)=0$. First, you find ${f}^{″}\left(x\right)$:
 ${f}^{″}\left(x\right)=60{x}^{2}$

Set this equal to 0 and you get: $\begin{array}{llll}\hfill 60{x}^{2}& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

You can now find the $y$-value of the inflection point by inserting your $x$-value into the main function $f\left(x\right)$:

 $f\left(0\right)=5{\left(0\right)}^{4}=0$

As you can see from this, the zero, the inflection point and the minimum are all the same point.

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