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
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
