How to tell if a function is increasing or decreasing from a derivative?

Solution:

Increasing functions are those functions that increase monotonically within a particular domain, and decreasing functions are those which decrease monotonically within a particular domain. We check for the monotonicity of a function using derivatives within the given domain.

Let's understand the statement in detail.

Let's take an example of a function f(x) = x³ + x² + 3x + 2. Let's find its monotonicity in the domain of real positive numbers.

First of all, we find its first derivative.

Hence, f'(x) = 3x² + 2x + 3.

Now since the function f'(x) is always positive for all the positive real numbers, it is said to be increasing in the domain of real positive numbers.

Hence, if the first derivative of a function is greater than zero in a particular interval, then it is said to be increasing in that interval, and vice-versa for decreasing function.

How to tell if a function is increasing or decreasing from a derivative?

Summary:

If the first derivative of a function is greater than zero in a particular interval, then it is said to be increasing in that interval, and vice-versa for decreasing function.