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) ex for all real numbers.
(ii) log x on (0, ∞)
Solution :
(i) f(x) = ex
Let f(x) = ex
f'(x) = ex
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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