Now, we learn from calculus that the slope of a function can be found by way of its derivative; what is more fascinating is that we can go further -much further.

Definition #1: Let be real valued on an interval . The following holds:

1) If whenever and are in with  , then our function is nondecreasing on I.

2) However, if  holds, then our function if strictly increasing.

The derivative plays a fundamental role in determining the nature of a functions monotonic behavior; we will show that the “slope” of our function on  can aid us in looking how it is moving along . We will prove this directly using the mean value theorem.

Theorem: let be differentiable on some .

i) If for all points in , then is non decreasing on .

ii) If for all points in , then is increasing on .

iii) If for all points in , then is non increasing on .

iv) If for all points in , then is decreasing on 

If for all points in , then is nondecreasing on 

Proof: To prove (i), let with . By the mean value theorem, so that:

So, when our derivative at c is positive or equal to zero, . If this holds on , then we say our function is nondecreasing on .

The absolutely awesomeness that spews from the prior fact is that we can use this to determine if there is a minimum or maximum at a point. If you imagine a function that is continuous on some real-valued interval , and, let that function have a minimum (or maximum) at a certain point in , then all you have to do is look at the values of the function from both sides as you reach that point. The function will hit a point where the derivatives have alternative signs -so, the value at that point will be bounded such that it must have a slope of zero. Thus, you have landed at a minimum.
Furthermore, consider that function . We know that  is not differentiable at x=0, but who cares? We can use this result to show that at the point x=0, the value of  changes so   goes from some specific type of behavior.

I would like to address a concept that is derived from those ideas. It is called Dini’s theorem -or Dini Derivatives.

We just discussed how we can “think” of the function  by way of our discussion of the values of its derivative. However, what other methods exist to look at this type of function more aggressively. I mean, one would think that there must be some method to evaluating the changes in the derivative itself even at points that may not have matching limits of the difference quotient at a specific point.

The graph above does not allow us to do much as we take the derivative near 0. In fact, a little work with the limit of the difference quotient while taking the limits as x goes to zero from the right and left will produce the obvious and one will witness their dream shattered and crushed under the weight of non-differentiability.

However, in mathematics, derivatives are IMPORTANT in evaluating the local behavior of a function. Will will discuss this in a subsequent post.