Question

Explain the concepts of increasing and decreasing functions ,using geometrical significance of dy/dx.

Answer 1

Step-by-step explanation:

Definition of an Increasing and Decreasing Function-:

• Let y=f(x) be a differentiable function on an interval (a,b). If for any two points x1,x2∈(a,b) such that x1<x2, there holds the inequality f(x1)≤f(x2), the function is called increasing (or non-decreasing) in this interval.

Answer 2

Explanation:

Let y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).

If for any two points x1 and x2 in the interval x such that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is called increasing in this interval.

Similarly, if for any two points x1 and x2 in the interval x such that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is called decreasing in this interval.

If a function is differentiable on the interval (a,b) and falls in any one of the four categories explained above i.e. Increasing/strictly Increasing, Decreasing/strictly decreasing, then the function is known as a Monotonic function. Note that if $\frac{df}{dx}$=0, the function is constant on that interval.

The First Derivative Test

This test can be used to find the nature of the function i.e. whether it is monotonic or not. It employs the use of the derivative of the function with respect to the independent variable at a point in the interval where the behaviour is to be determined. For a function f(x) in the domain (a,b):

If dfdx≥0 for all x in (a,b), then f(x) is an Increasing Function in (a,b).

Similarly,

If dfdx≤0 for all x in (a,b), then f(x) is an Decreasing Function in (a,b).

For the strictly monotonic nature, strict inequalities must hold i.e.

dfdx>0 for Strictly Increasing Functions

dfdx<0 for Strictly Decreasing Functions