Increasing Function





In this page increasing function we are going to see how to check the given function is increasing or not. A function f is said to be increasing function on an interval I if f (x₁) ≤ f (x₂) when x₁ < x₂ in I.

How to find whether the given function is increasing in the given interval.

Step 1: Find the first derivative  

Step 2: Apply random values from the given interval. If the simplified value be positive then we can say the given function is increasing. If it is negative we can say the given function is decreasing.

Example 1:

Check whether y = x² is increasing on the interval (0,∞)

As per the procedure first let us find the first derivative.

           dy/dx = 2 x

            f '(x) = 2x

Now let us apply random values from the given interval

x = 2 ∈ (0,∞)

            f '(2) = 2(2)

                     = 4 > 0

x = 6 ∈ (0,∞)

            f '(6) = 2(6)

                     = 12 > 0

So the given function is increasing in the interval (0,∞).

increasing function


Example 2:    

Check whether y = sin x is increasing on the interval (0,Π/2)

As per the procedure first let us find the first derivative

           dy/dx = cos x 

            f '(x) = cos x 

Now let us apply random values from the given interval

x = Π/3 ∈ (0,Π/2)

            f '(Π/3) = cos Π/3

                       = 1/2 > 0

x = Π/6 ∈ (0,Π/2)

            f '(Π/6) = cos Π/6

                       = √3/2 > 0  

So the given function is increasing in the interval (0,Π/2).


Example 3:

Check whether y = 2x³ + x² - 20 x is increasing on the interval (-∞,-2)

As per the procedure first let us find the first derivative

           dy/dx = 2 (3x²) + 2x - 0

           dy/dx = 6x² + 2x

            f '(x) = 6x² + 2x

Now let us apply random values from the given interval

x = -5 ∈ (-∞,-2)

            f '(-5) = 6(-5)² + 2(-5)

                     = 6(25) -10

                     = 150 - 10

                     = 140 > 0  Increasing Function

x = -3 ∈ (-∞,-2)

            f '(-3) = 6(-3)² + 2(-3)

                      = 6(9) - 6

                      =  54 - 6

                      =  48 > 0

So the given function is increasing in the interval (-∞,-2).

Related Topics










Increasing Function to Decreasing Function