How to Find the Increasing or Decreasing Functions?

Increasing and decreasing functions are functions in calculus for which the value of \(f(x)\) increases and decreases respectively with the increase in the value of \(x\).

How to Find the Increasing or Decreasing Functions?

If the value of \(f(x)\) increases with the increasing value of \(x\), the function is said to be increasing, and if the value of \(f(x)\) decreases with the increasing value of \(x\), the function is decreasing.

Step by Step guide to increasing and decreasing functions

Increasing and decreasing functions are functions whose graphs go up and down respectively by moving to the right of the \(x\)-axis. Increasing and decreasing functions are also called non-decreasing and non-increasing functions.

Increasing and decreasing functions definition

Increasing function: The function \(f(x)\) in the interval \(I\) is increasing on anif for any two numbers \(x\) and \(y\) in \(I\) such that \(x<y\), we have \(f(x) ≤ f(y)\).

Decreasing function: The function \(f(x)\) in the interval \(I\) is decreasing if for any two numbers \(x\) and \(y\) in \(I\) such that \(x<y\), we have \(f(x) ≥ f(y)\).

Strictly increasing function: A function \(f(x)\) is called to be strictly increasing on an interval \(I\) if for any two numbers \(x\) and \(y\) in \(I\) such that \(x<y\), we have \(f(x) < f(y)\).

Strictly decreasing function: A function \(f(x)\) is called to be strictly decreasing on an interval \(I\) if for any two numbers \(x\) and \(y\) in \(I\) such that \(x<y\), we have \(f(x)>f(y)\).

Graphical representation of increasing and decreasing functions

Graphical representation of increase and decrease functions helps us understand the behavior of the functions.

As we can see above in the diagrams, the increasing function includes both strictly increasing intervals and intervals where the function is constant. Similarly, a decreasing function contains intervals where the function is strictly decreasing and where the function is constant.

Rules to check increasing and decreasing functions

We use a derivative of a function to check whether the function is increasing or decreasing. Suppose a function \(f(x)\) is differentiable on an open interval \(I\), then we have:

  • If \(f'(x) ≥ 0\) on \(I\), the function is said to be an increasing function on \(I\).
  • If \(f'(x)≤ 0\) on \(I\), the function is said to be a decreasing function on \(I\).

Note: The first derivative of a function is used to check for increasing and decreasing functions.

Properties of increasing and decreasing functions

  • If the functions \(f\) and \(g\) are increasing functions on an open interval \(I\), then the sum of the functions \(f+g\) is also increasing on this interval.
  • If the functions \(f\) and \(g\) are decreasing functions on an open interval \(I\), then the sum of the functions \(f+g\) is also decreasing on this interval.
  • If the function \(f\) is an increasing function on an open interval \(I\), then the opposite function \(-f\) decreases on this interval.
  • If the function \(f\) is a decreasing function on an open interval \(I\), then the opposite function \(-f\) is increasing on this interval.
  • If the function \(f\) is an increasing function on an open interval \(I\), then the inverse function \(\frac{1}{f}\) is decreasing on this interval.
  • If the function \(f\) is a decreasing function on an open interval \(I\), then the inverse function \(\frac{1}{f}\) is increasing on this interval.
  • If the functions \(f\) and \(g\) are increasing functions on an open interval \(I\) and \(f, g ≥ 0\) on \(I\), then the product of the functions \(fg\) is also increasing on this interval.
  • If the functions \(f\) and \(g\) are decreasing functions on an open interval \(I\) and \(f, g ≥ 0\) on \(I\), then the product of the functions \(fg\) is also decreasing on this interval.

Increasing and Decreasing Functions – Example 1:

Determine the interval(s) on which \(f(x)=xe^{-x}\) is increasing. 

Solution:

First, find the derivative of \(f(x)\).

\(f(x) = xe^{-x}\)

\(f'(x) = e^{-x} – xe^{-x}\)

\(= e^{-x}(1 – x)\)

To determine the critical point, equate \(f'(x)\) with \(0\), that is, \(e^{-x}(1 – x) = 0 ⇒ x = 1\) (because exponential function can not be equal to \(0\)).

  • For \(x< 1, (1 – x) > 0 ⇒e^{-x}(1 – x)>0\) (because exponential is always positive)
  • For \(x > 1, (1 – x) < 0 ⇒ e^{-x}(1 – x)<0\) (because exponential is always positive)

Now, we have \(f'(x)>0\) for \(x<1\). Therefore, the interval where \(f(x)=xe^{-x} \) is increasing on \((-∞, 1)\).

Exercises for Increasing and Decreasing Functions

Determine the intervals at which the function is increasing.

  • \(\color{blue}{f\left(x\right)=x\:ln\:x}\)
  • \(\color{blue}{f\left(x\right)=4x-x^2}\)

Determine the intervals at which the function is decreasing.

  • \(\color{blue}{f\left(x\right)=5-2x-x^2}\)
  • \(\color{blue}{f\left(x\right)=xe^{3x}}\)
This image has an empty alt attribute; its file name is answers.png
  • \(\color{blue}{(\frac{1}{e}, \infty)}\)
  • \(\color{blue}{(-\infty,2)}\)
  • \(\color{blue}{(-1,\infty)}\)
  • \(\color{blue}{\left(-\infty ,-\frac{1}{3}\right)}\)

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