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# What are the Increasing and decreasing functions?

Introduction(Increasing and decreasing functions)- A function is said to be increasing function in an interval [a, b] if y increases as well as x increases from a to b.

Or in other words, A function is said to be an increasing function in [a, b] if its first derivative is greater than zero for all values of x in an interval [a, b].

And

A function is said to be decreasing function in an interval [a, b] if y decreases as well as x increases from a to b.

“If the first derivative is less than zero for all values of x in an interval [a, b] then the function y = f(x) is a decreasing function in [a, b].

## Maximum and minimum(Increasing and decreasing functions)

We call a function f(x) is maximum at x = a. If f(a) is greater than every other value of f(x) in the immediate neighbourhood of x = a. Then (i.e., f(x) ceases to increase but begins to increase at x = a.

Similarly, the minimum value of f(x) will be that value at x = b which is less than other values in the immediate neighbourhood of x = b.

## Maxima and minima of function of two variables

As we know that the value of a function at maximum point is the maximum value of a function.

Similarly, the value of a function at minimum point is the minimum value of a function.

The maxima and minima of a function is an extreme biggest and extreme smallest point of a function in a given range (interval) or entire region.

Pierre de Fermat was the first mathematician to discover general method for calculating maxima and minima of a function. The maxima and minima are complement of each other.

## Maxima and Minima of a function of one variables

If  f(x) is a single value function in a region R, then-

Maxima is a maximum point  if and only if

Minima is a minimum point  if and only if