What is it?
One of the most popular mathematical applications for Differentiation is optimization, which translates to the most optimal way of doing something. Commonly, these problems can be reduced to finding the maximum and minimum values of a function.
The concept of finding the maximum and minimum of a function is widely used in Machine Learning and Gradient Descent methods, to minimize the error of a given algorithm.
Minimum and maximum values
The absolute maximum/ minimum value of a function, also called global maximum/ minimum value, is the highest possible value of a function over its entire domain. So the function cannot output any other value higher/ lower than it. These points are called the extreme values of a function.
But there is also the local maximum/ minimum value, which is the highest/ lowest value but only within a specific region or interval. So one may mistake a local maximum and treat as the global, but it may not be true.
Some functions may or not present extreme values, when its values can go infinitely lower or higher. A function also need to be continuous to have an extreme value. For example, the function has no local or global extreme values.
The Fermat Theorem
Pierre Fermat was a French mathematician that founded analytic geometry with Descartes, and part of his importance is for finding tangents to curves and extreme values of functions before the invention of limits and derivative.
Fermat proposed a theorem, called the Fermat’s Theorem, which states:
If has a local maximum or minimum at , and if exists, then .
Even though there are exceptions, for example, , Fermat’s Theorem suggests that we could start searching these values at , or where does not exist.
These values at these points are called critical numbers.
The Closed Interval method
We can then use these critical numbers to find the local extreme values of a continuous function on a closed interval .
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Find the values of at the critical numbers of in .
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Find the values of at the endpoints of the interval.
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The largest/ smallest value found is the local extreme value.
To test the this value, one can use the First Derivative Test.
Inflection points
The point of inflection, or inflection point, describe the change in movement from a function , where the concavity switches. The point where the concavity of a function goes from one direction to another is called the inflection point.
All inflection points have a second derivative of 0, if it exists, but not all points with a second derivative of 0 are inflection points.
Finding the inflection point
The process to find the inflection point is very similar to finding the extreme values of a function. Given a continuous function :
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Find the second derivative of , .
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Solve it for and for x-values where is undefined, similar to the critical numbers.
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Plug the value back to the original formula and get the y-values.
You can also test this values similar to the First Derivative Test, but use the second derivative, for testing.
With multiple inflection points you can plot a sign graph and determine the intervals at which the concavity of a function would be up or down. Given , the plot and sign graph would be:
