Monotonic Function – Definition and Examples

Here you will learn definition of monotonic function and condition for monotonicity with examples.

Let’s begin –

Monotonic Function

The function f(x) is said to be monotonic on an interval (a, b) if it is either increasing or decreasing on (a, b).

A function f(x) is said to be increasing (decreasing) at a point \(x_0\), if there is an interval (\(x_0 – h, x_0 + h\)) containing \(x_0\) such that f(x) is increasing (decreasing) on (\(x_0 – h, x_0 + h\)).

A function f(x) is said to be increasing (decreasing) on [a, b] if it is increasing (decreasing) on (a, b) and it is also increasing (decreasing) at x = a and x = b. 

Necessary and Sufficient Conditions for Monotonicity

Let f be a differentiable real function defined on an open interval (a, b).

(i) If f'(x) > 0 for all x \(\in\) (a, b), then f(x) is increasing on (a, b).

(ii) If f'(x) < 0 for all x \(\in\) (a, b), then f(x) is decreasing on (a, b).

Corollary : 

Let f(x) be a function defined on (a, b).

(i) If f'(x) > 0 for all x \(\in\) (a,b) except for a finite number of points, where f'(x) = 0, then f(x) is increasing on (a, b).

(ii) If f'(x) < 0 for all x \(\in\) (a,b) except for a finite number of points, where f'(x) = 0, then f(x) is decreasing on (a, b).

Algorithm

1 : Obtain the function and put it equal to f(x).

2 : find f'(x)

3 : Put f'(x) > 0 and solve this inequation.

for the values of x obtained in step 3 f(x) is increasing and for the remaining points in its domain it is decreasing.

Example : find the interval in which f(x) = \(-x^2 – 2x + 15\) is increasing or decreasing.

Solution : We have, 

f(x) = \(-x^2 – 2x + 15\)

\(\implies\) f'(x) = -2x – 2 = -2(x + 1)

for f(x) to be increasing, we must have

f'(x) > 0

-2(x + 1) > 0

\(\implies\) x + 1 < 0

\(\implies\) x < -1 \(\implies\) x \(\in\) \((-\infty, -1)\).

Thus f(x) is increasing on the interval \((-\infty, -1)\).

for f(x) to be decreasing, we must have

f'(x) > 0

-2(x + 1) < 0

\(\implies\) x + 1 > 0

\(\implies\) x > -1 \(\implies\) x \(\in\) \((-1, \infty)\).

Thus f(x) is decreasing on the interval \((-1, \infty)\).


Related Questions

Prove that \(f(\theta)\) = \({4sin \theta\over 2 + cos\theta} – \theta\) is an increasing function of \(\theta\) in \([0, {\pi\over 2}]\).

Separate \([0, {\pi\over 2}]\) into subintervals in which f(x) = sin 3x is increasing or decreasing.

Prove that the function f(x) = \(x^3 – 3x^2 + 3x – 100\) is increasing on R

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