Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions

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Lesson Summary: 
We show how the first derivative can be used to identify when a function is either increasing or decreasing on an interval.
Lesson Inputs: 
The Slope Of A Straight Line
Geometry of Functions I: Increasing and Decreasing Functions
Geometry of Functions III: The Mean-Value Theorem
Lesson Outputs: 
Geometry of Functions V: Maximum and Minimum

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Lesson: 

Recall our lesson on increasing and decreasing functions. For an increasing function <Sign in to see all the formulas> in an interval $I$. Therefore, the secant line between <Sign in to see all the formulas> and <Sign in to see all the formulas> has a slope of

<Sign in to see all the formulas>


on the interval $I$. It is important to note that for an increasing function the slope is always positive for an increasing function. Similarly, for a decreasing function <Sign in to see all the formulas> and the slope is always negative. This suggests the idea that for increasing functions the slope in the interval $I$ is always positive and for a decreasing function it is always negative. What we will do is present two scenarios. The first is when the interval is an open interval. The second is when the interval is a closed interval.

Theorem 1
Assume that $f(x)$ is differentiable on an open interval $I$. In this case we know that

Theorem 1 above has a small problem. For example, the function <Sign in to see all the formulas> has the derivative <Sign in to see all the formulas> which is negative in the interval <Sign in to see all the formulas>, zero at $x=0$ and positive in the interval <Sign in to see all the formulas>. From the theorem above this means that <Sign in to see all the formulas> decreases on <Sign in to see all the formulas> and increases on <Sign in to see all the formulas>. However, we know that it actually decreases on <Sign in to see all the formulas> and increases on <Sign in to see all the formulas>. The point is that the theorem above does not handle the case where we have closed intervals. We handle the case where we have closed intervals by adding the requirement of continuity at the endpoints. Recall that it is already assumed to be continuous in the open interval because its derivative is assumed to exist.

Theorem 2
Assume that $f(x)$ is continuous on a closed interval $I$ and differentiable on the interior of $I$. In this case we know that

A proof of Theorem 1 may be found here and a proof of Theorem 2 may be found here. While we don't recommend learning the proofs if you are not required to, the proofs are relatively simple and utilize the very important Mean-Value Theorem.