Functions
Power functions are a special case of polynomial functions. Power functions only have one term—a polynomial in the form .
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
- 1.
- First, you find the zeros by setting . You then get:
So you have a zero at .
- 2.
- You then find the maxima and minima by setting . First, you differentiate the function:
You then set this equal to 0 to find the maxima and minima:
To determine if it is a maximum or a minimum, you can select two values, one to the left of and one to the right of . Input these into the differentiated function and interpret the sign. Choose numbers that are easy to work with, like and :
As the graph decreases first and then increases, this is a minimum.You now need to find the -value of the point, by inserting the -value into the main function :
Thus, you have a minimum at .
- 3.
- To find the inflection points of the function, you set . First, you find :
Set this equal to 0 and you get:
You can now find the -value of the inflection point by inserting your -value into the main function :
As you can see from this, the zero, the inflection point and the minimum are all the same point.