Critical Point Calculator

A critical point calculator finds the critical point for one or two variable functions and saddle points for multivariable functions. Take the help of our critical point finder to deal with the local minima and maxima for a given function. 

What are Critical Points?

“A critical point of a continuous function is a point at which the derivative is either zero or undefined where the function is not differentiable”. 

Critical Points on Functions:

The value of a function at a critical point is a critical value. In the graph, these values are placed where the function is zero because their function's rate of change is altered by either a change from increasing to decreasing. 

In order to find the critical numbers of the function there is a simple procedure:

• Take its derivatives 
• Set these equal to the zero 
• And solve for x 

Any x value that makes a derivative zero are critical number. The critical point of the multidimensional function is the point where the first-order partial derivative of a function is zero. 

Evaluation Process of Critical Variables:

An online critical number calculator is designed to deal with the critical points for one, two, or multivariable functions. Do you know that these points are calculated by putting the first derivative equal to the zero f'(x) = 0? 

Let us examine the examples below that describe and clarify how to find critical points in a better way. 

One Variable Example:

Suppose a single variable (x^2 + 4x + 9) is available to find those points where the functions are not differentiable. 

Solution:

Step # 1:

Derivative Steps:

$$ \frac{\partial}{\partial x}\left(x^{2} + 4 x + 9\right) $$

Differentiate $$ x^{2} + 4 x + 9 $$ term by term:

Step # 2:

Apply the power rule:

Where x^2 goes to 2x and the result is 4 

The critical points calculator utilizes the derivative of constant times a function and once again applies the power rule in which x goes to 1.

The derivative of constant 9 is zero so behind this we have 2x + 4 = 0 

How to find critical numbers simply put f'(x) = 0

$$ 2 x + 4 = 0 $$

Local Minima:

$$ (x,f(x))=\text{No local minima} $$

Local Maxima:

$$ (x,f(x))=(-2 , 5.0) $$

Root = [-2]

Two Variable Example: 

2x^2 + 5xy +2y

Solution:

Step # 1:

$$ \frac{\partial}{\partial x}\left(2 x^{2} + 5 x y + 2 y\right) $$

Differentiate term by term by using a critical points calculator so the derivative of a constant times a function is the constant times the derivative of the function.

Step # 2:

Apply the power rule

Where 

• x^2 goes to 2x and result is 4x
• x goes to 1 and the result is 5y
• Constant 2y goes equal to zero

The derivative of a constant times a function is the constant times the derivative of the function. So the conclusion is 4x + 5y 

Step # 3:

$$ \frac{\partial}{\partial y}\left(2 x^{2} + 5 x y + 2 y\right) $$

The multivariable critical point calculator applies a power rule and we get 

• 2x = 0, 
• y = 1 and the result is 5x 
• y goes to 1 and the result is 2 

The derivative of a constant times a function is the constant times the derivative of the function. So the conclusion is 5x + 2 = 0

For finding critical points of a function put f'(x,y) = 0

$$ 4 x + 5 y = 0 $$

$$ 5 x + 2 = 0 $$

Critical points = $$ \left\{ x : - \frac{2}{5}, \  y : \frac{8}{25}\right\} $$

Working of Critical Point Calculator:

The critical number calculator determines those points on which the function is not differentiable. This critical point finder streamlines the process of identifying the critical points within seconds. 

Input:

• Enter the expression to find critical points
• Hit on “Calculate” icon 

Output:

Below are the results you will receive when using the critical point calculator.

• Critical points of a given function
• Local Minima and Maxima 
• Calculations in steps by using the power rule

Frequently Asked Questions:

Why is Zero not a Critical Point?

The point in the domain of the function is undefined or zero is referred to as critical point. So x is equal to zero and will not count as a critical point if it is not in the original function. 

Do Critical Points Include Endpoints?

Yes, it includes the endpoints. As we know it is a function of the domain where f’(x) is equal to zero. It can also be the end point of an interval. 

References:

From the source Wikipedia: Critical point, Critical point of a single variable function, Critical points of an implicit curve, Critical point of a differentiable map, Application to topology. 

From the source Khan Academy: Critical point introduction, Find critical points for variables.