Maxima and Minima is a very important concept of calculus. We can use this concept in real life scenario whenever we want to make optimal usage of available resources so that profit is maximized.
Explanation To understand the concept with clarity it’s important to have basic idea of derivative, Increasing and Decreasing functions. Derivative in layman terms is nothing but change. In mathematical terms it represents slope of a function at the point of interest. Slope of a function is given by tangent drawn at the point of interest.
An Increasing function is one whose slope is non-negative. If the slope is strictly positive, it is known as monotonously increasing function.
A decreasing function is one whose slope is non-positive. If the slope is strictly negative, it is known as monotonously decreasing function.
In Maxima and Minima, we learn about both Local and Global Maximum and Minimum Points.
For Local Maxima and Minima, the point should satisfy the condition that, the first derivative of function equals to zero. For Maxima, second derivative is less than zero and for Minima, second derivative is greater than zero. If the second derivative is equal to zero, it is known as point of inflection. The points where maxima and minima happen are known as critical or extreme points.
For an nth degree polynomial, number of extrema will not exceed n-1 and zero crossing cannot exceed.
For Global Maxima we should find the maximum value of all extreme points also taking in to consideration the function value at end points. And same goes for Global minima by considering Minimum value points including initial and final points.