Monotonic Function

In this page monotonic function we are going to discuss about what is monotonic-function and its examples.


If a function changes its sign at different points on the interval then the function is said to be not monotonic.

Procedure to check whether the given function is monotonic:

  • Find the first derivative.
  • Check different points on the given interval
  • If we get different signs, we can decide the given function is monotonic. Otherwise it is not monotonic.

Example 1:

Check whether the function y = sin x + cos 2x is monotonic on the interval [0,Π/4]


                    f (x) = sin x + cos 2x

                   f '(x) = cos x - sin 2x [2 (1)]

                   f '(x) = cos x - 2 sin 2x

Now let us apply x = 0

                   f '(0) = cos 0 - 2 sin 2(0)

                   f '(0) = 1 - 2 (0)

                   f '(0) = 1 - 0

                   f '(0) = 1 > 0

then apply x = Π/4

                   f '(Π/4) = cos Π/4 - 2 sin 2(Π/4)

                   f '(Π/4) = 1/√2 - 2 sin (Π/2)

                   f '(Π/4) = 1/√2 - 2 (1)

                   f '(Π/4) = 1/√2 - 2

                   f '(Π/4) = 0.707 - 2

                   f '(Π/4) = -1.292 < 0

The given function is not monotonic function on the interval [0,Π/4]

Example 2:

Check whether the function y = x sin x is monotonic on the interval [Π/2,Π]


                    f (x) = x sin x

                   f '(x) = x (cos x) + sin x (1)

                   f '(x) = x (cos x) + sin x

Now let us apply x = Π/2

                   f '(0) = Π/2 (cos Π/2) + sin Π/2

                   f '(0) = Π/2 (0) + 1

                   f '(0) = 1 > 0

then apply x = Π

                   f '(Π) = Π (cos Π) + sin Π

                   f '(0) = Π (-1)  + 0

                   f '(0) = - Π

Related Topics

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

Monotonic Function to Decreasing Function

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