MONOTONICITY OF THE FUNCTION

Definition :

A function that is completely increasing or completely decreasing on the given interval is called monotonic on the given interval.

Test for monotonic functions :

Let I be an open interval. Let f : I → R be differentiable. Then

(i) The function f is increasing if and only if f ′(x) ≥ 0 for all x in I.

(ii) The function f is decreasing if and only if f ′(x) ≤ 0 for all x in I.

Example 1 :

Check whether the function

y = sin x + cos 2x

is monotonic on the interval [0,Π/4]

Solution :

f(x)  =  sin x + cos 2x

f'(x)  =  cos x - sin 2x [2 (1)]

f'(x)  =  cos x - 2 sin 2x

By applying  x = 0, we get

f'(0)  =  cos 0 - 2 sin 2(0)

f'(0)  =  1-2 (0)

f'(0)  =  1 - 0

f'(0)  =  1 > 0

By applying x = Π/4, we get

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

Thus f′ is of different signs at 0 and π/4. So, 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]

Solution :

f(x)  =  x sin x

f'(x)  =  x (cos x) + sin x (1)

f'(x)  =   x (cos x) + sin x 

By applying x = Π/2

f'(0)  =  Π/2 (cos Π/2) + sin Π/2

f'(0)  =  Π/2 (0) + 1

f'(0)  =  1 > 0

By applying x = Π

f'(Π)  =  Π (cos Π) + sin Π

f'(0)  =  Π (-1)  + 0

f'(0)  =   - Π < 0

So, the function is not monotonic function.

Example 3 :

Discuss monotonicity of the function

(i)  e^x for all real numbers.

(ii) log x on (0, ∞)

Solution :

(i)  f(x)  =  e^x 

Let f(x)  =  e^x

f'(x)  =  e^x

• If x > 0, then f′(x) > 0. The function is strictly increasing for all positive values of x.
• If x < 0 then f′(x) > 0. The function is strictly increasing for all negative values of x.
• If x  =  0, then f'(x)  =  1,we get positive value for f'(x).

So, the function monotonic on real numbers.

(ii)  f(x)  =  log x on (0, ∞)

Let f(x)  =  log x

f'(x)  =  1/x

If x > 0, then f′(x) > 0. The function is strictly increasing for all positive values of x.

So, the function monotonic on (0, ∞).

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