Monotonicity Of A Function

What Is Monotonicity of a function

Monotonicity is an inherent characteristic or property of a function with the help of which, we can track the behaviour and patterns exhibited by any function graph. The monotonically meaning of any function or an equation signifies that if the graph is increasing with increasing values of x coordinate, then the graph will be called monotonically increasing [if x₁ < x₂ & f(x₁) ≤ f(x₂)]. Similarly, if the range value of the graph is decreasing with the increasing values of x coordinate, then the graph will be declared monotonically decreasing [if x₁ < x₂ & f(x₁) ≥ f(x₂)]. 

y = 3x + 5, y = ex , y = log(x), are the examples of monotonically increasing function and y = (-x)6 and y = e-x are the examples of monotonically decreasing function.

How To Check Monotonic function

1. Monotonicity check at one point: – At any point on the graph of a monotonically increasing or monotonically decreasing function can be checked by the help of drawing a tangent to that point and observing whether that tangent is making an acute angle or an obtuse angle with the x-axis. If the tangent is making an acute angle then it means that the function is monotonically increasing at that point, and if the tangent is making an obtuse angle with the x-axis then it means that the function is monotonically decreasing at that point. 
2. Monotonicity for an interval: – Let p, and q be the two-interval points and f(x) be the function.

The function is monotonically increasing if first derivative of f(x),f1(x) >= 0.

The function is monotonically decreasing if first derivative of f(x),f1(x) <= 0.

The function is monotonically constant if first derivative of f(x),f1(x) = 0.

Extremum In A Function

Extremum: – Extremum is a spot on the graph of any valid function where either there is a maximum value for the function lies or the minimum value for the function exists. If you’ll think further, you’ll understand that extremum points are located at the domain values ad those points are the places where a function changes its direction from top to bottom or bottom to top. The derivative value of that function at that point will be zero. 

Three simple cases under extremum of a function: –

1. The function f(x) has to be a monotonically increasing function in between particular interval points, a, and b. Then, f(a) will give the least value, whereas f(b) will give the maximum value.
2. The function f(x) has to be a monotonically decreasing function in between particular interval points, a, and b. Then, f(a) will give the maximum value, whereas f(b) will give the minimum value.
3. For the function to become a non-monotonic function in between the interval [a,b], the minimum point and the maximum point will lie at those places where df(x)/dx = 0.

Maxima And Minima

Local Maxima is the point on the graph or the curve where the value of the function is higher than the limiting function value.

Local Minima is the point on the graph or the curve where the value of the function is lower than the limiting function value.

Global Maxima is the point on the graph or curve that is the maximum and the highest value of the function among different numbers of critical points in the function.

Global Minima is the point on the graph or curve that is the minimum and least value of the function among different numbers of critical points in the function.

A function f(x) is said to obtain a maximum at x = p if there exists a neighbouring point within the close proximity of other x elements, so that (p – θ, p + θ). 

Some Solved Problems

Q1. Prove that y =  f(x) = x – cos(x) is an increasing function by the help of monotonicity. 

Solution: f(x) = x – cos(x)

=> dy/dx = 1 + sin(x)

dy/dx will always be greater than zero because sin(x) having values within the interval [-1,1] and dy/dx = 0, hence we can include this function in a monotonically increasing function.

Conclusion

A monotonic function does help in aiding the simplification to get to the depth of limits and neighbouring elements. A function can have a multiple number of local maxima and local minima at multiple points but there is only a single global maximum or single global minimum. The value of these local maxima and local minima does not necessarily has to do anything with the global maxima and global maximum.  This article showed us the importance of the monotonicity of a function, its importance at the time of studying the continuity and derivatives of the functions.