Loading....
Coupon Accepted Successfully!

 

Graph of Non-Monotonic Function

88676.png

Increasing function on interval

f(x) is increasing in interval (a, b) if for all x1, x2 (a, b), if x1 < x2 f(x1) ≤ f(x2).
 
88670.png
 
In the given graph of function if we draw any tangent line, it makes either acute angle with positive direction of the x-axis or it is parallel to the x-axis, i.e. slope of tangent at any point is non-negative. Also slope of tangent is equal to derivative of function at that point, i.e., f′(x) ≥ 0 for all xin the interval (a, b). So when f′(x) ≥ 0 for all x in the interval (a, b) of consideration, f(x) is said to be increasing.
 
For example, for f(x) = x3, f′(x) = 3x2 ≥ 0, i.e. f(x) is an increasing function on R (domain of function).

Strictly increasing function on interval

f(x) is strictly increasing in interval (a, b) if for all x1, x2 (a, b), if x1 < x2 f(x1) < f(x2).
In the given graph of function if we draw any tangent line, it makes acute angle with positive direction of x-axis, i.e., slope of tangent at any point is positive, i.e., f′(x) > 0 for all x in the interval (a, b). So when f′(x) > 0 for all x in the interval (a, b) of consideration, f(x) is said to be strictly increasing.
 
89396.png
 
For example, for f(x) = log x, f′(x) = 1/x > 0 (for x > 0), i.e., f(x) is strictly increasing function on (0, ∞) (domain of function).

Decreasing function on interval

f(x) is decreasing in interval (a, b) if for all x1, x2 (a, b), if x1 < x2 f(x1) ≥ f(x2).
 
88657.png
 
In the given graph of function if we draw any tangent line, it makes either obtuse angle with positive direction of the x-axis or it is parallel to the x-axis, i.e., slope of tangent at any point is non-positive, i.e., f ′(x) ≤ 0 for all x in the interval (a, b). So when f′(x) ≤ 0 for all x in the interval (a,b) of consideration, f(x) is said to be increasing.
 
For example, for f(x) = sin xx, f′(x) = cos x – 1 ≤ 0 (as – 1 ≤ cos x ≤ 1), i.e. f(x) is a decreasing function on R (domain of function).

Strictly decreasing function on interval

f(x) is strictly decreasing in interval (a, b) if for all x1, x2 (a, b), if x1 < x2 f(x1) > f(x2)
In the given graph of function if we draw any tangent line, it makes obtuse angle with positive direction of the x-axis, i.e., slope of tangent at any point is positive, i.e., f′(x) < 0 for all x in the interval (a, b). So when f′(x) < 0 for all x in the interval (a, b) of consideration, f(x) is said to be strictly decreasing.
 
89402.png
 
For example, for f(x) = ex, f′(x) = – ex < 0 (as ex > 0 for all real x), i.e., f(x) is strictly decreasing function on R (domain of function).

Increasing/decreasing at a point

Let x0 be a point in the domain of definition of a real- valued function f. Then f is said to be increasing, strictly increasing, decreasing or strictly decreasing at x0 if there exists an open interval I containing x0 such that f is increasing, strictly increasing, decreasing or strictly decreasing, respectively, in I.
 
For example, a function f is said to be increasing at x0 if there exists an interval I = (x0h, x0 + h), h > 0 such that for x1, x2 I and x1 < x2 in I f(x1) ≤ f(x2).




Test Your Skills Now!
Take a Quiz now
Reviewer Name