Maxima and Minima
“The art of doing mathematics is finding that special case that contains all the germs of generality” – David Hilbert
In this article, we are going to learn about the concept of Maxima and Minima which has a significant role in Deep Learning. We will also learn about slopes and use them to find the maxima and minima of the function.
Let’s assume a function as shown in figure below:
Let’s take 2 points and on and let’s observe the situation if we change the input then how the value and of function will change, we can observe the situation in the below animation:
We can clearly see that if we increase the input, the output is also increasing, so let’s assume and . Now,
and we know that OR we can say that is positive OR the slope of function where is positve, as the function changing positvely on positive nudge on input.
Let’s observe another situation on a different function f_1(k) as shown below:
Again, let’s the same previous situation on this function by taking two points on , and let’s observe the rate of change of this function :
We can notice that if we increase the input, the output is decreasing, so let’s and . Now,
as , we can say that is negative OR the slope of function where is negative, as the function is decreasing on positive nudge on input.
There are also some points when OR the point where function doesn’t change by nudging the input and stays at constant value as OR the derivative is . In function , let’s zoom to a specific portion and let’s observe the :
We can observe that, while nudging the input, function almost stays the same at that portion and doesn’t change at all. The point where and will certainly have a Maxima or a Minima. We can say that point(s) where the derivative/slope of a function becomes will definitely have either Maxima or Minima.
Let’s take an example and find the points where the derivative of that function is and will visualize that if at that point there exists Maxima/Minima:
we know that, for determining maxima/minima, the derivative of the function should be at some point
The values of where are , so at and at there should be maxima/minima. You can observe the figure below:
So, let’s conclude this article here and in next article we will explore the methods through which we can find where the point of Maxima and Minima exists for a function and will also explore the topic where the derivative of the function is not defined.