>Applications of the Derivative>How to Analyze Power Functions

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

Analyzing Power Functions

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

Example 1

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

1.

First, you find the zeros by setting

f (x) = 0

. You then get:

\begin{array}{l} 5 x^{4} & = 0 | \div 5 \\ x^{4} & = 0 \\ \sqrt[4]{x^{4}} & = \sqrt[4]{0} \\ x & = 0 \end{array}

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

2.

You then find the maxima and minima by setting

f^{'} (x) = 0

. First, you differentiate the function:

f^{'} (x) = 20 x^{3}

You then set this equal to 0 to find the maxima and minima:

\begin{array}{l} 20 x^{3} & = 0 \\ x & = 0 \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 $f^{'} (x)$ and interpret the sign. Choose numbers that are easy to work with, like $x = - 1$ and $x = 1$ :