Geometry of Functions VI: Critical Points
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Now let's put a few things together. At this point you should be very comfortable with the concept that a derivative is the slope of a function at a specific point. Furthermore, we have shown that functions are increasing when their derivatives are positive and decreasing when their derivatives are negative.
So now let's look at a function that has a maximum. In the picture below we will label when the function is increasing and when it is decreasing. We will also label when the slope is positive and when it is negative. Take a look:
Similarly, we can look at a function that has a minimum:
The pictures suggest that a function will have a slope of zero where it peaks for a maximum and where it bottoms out for a minimum. We call a point on a function where its derivative is zero or where its derivative does not exist a critical point. The relationship between critical points and maxima and minima is so important that we have a theorem about it:
| Critical Point Theorem |
|---|
| If $f(x)$ has a maximum or minimum at $e$, then <Sign in to see all the formulas> or <Sign in to see all the formulas> does not exist. |
In other words, if there is a maximum or minimum at $e$, then $e$ is a critical point. You may find a proof of the theorem here.
There are two more important things to note. The first is that maxima and minima exist at critical points, but not all critical points have a maximum or minimum. The easiest example of this is <Sign in to see all the formulas>. It has a critical point at $x=0$, but by graphing it you can see that $x=0$ is not a maximum or minimum.
The second thing to note is that the derivative does not have to exist at a maximum or minimum. We mention this above in the Critical Point Theorem, but an example works best to show what we mean. For example, we know that <Sign in to see all the formulas> does not have a derivative at $x=0$, but we can use our Function Wizard to graph it and see that there is a minimum there:
