Critical Points Calculator

Critical point calculator

A critical point is a function point at which the function's differential is undefined or zero. It can be characterized as a point on the function graph where the differentiation is infinite or zero. 

Critical points can be determined by putting the first derivative equivalent to zero f’(x) = 0.

How to use our Critical Point Calculator:

Here are the steps to calculate the Critical points of the function:

Step 1: Input the function:

First, enter the function you want to calculate. Such as f(x) = 3x^2 + xy +2y.

Step 2: Submit:

Click the submit button to get your output.

How to compute Critical Points?

To figure out how to compute the critical point, follow the below examples 

Example 1: 4x^2 + 4x + 3

1: Compute derivative w.r.t x: 

    = ∂/∂x (4x^2 + 4x + 3) 

    = 8x + 4 

2: Now to find critical points put the first derivative equal to zero: 

     8x + 4 = 0 

     8x = -4 

     x = -48 = -12

 Hence the critical point is  -12

Example 2: 3x^2 + xy + 2y 

1: Compute first derivative w.r.t x: 

     = ∂/∂x (3x^2 + xy + 2y)

     = 6x + y         --→ eq 1 

2: Now again compute derivative w.r.t y: 

     = ∂/∂y (3x^2 + xy + 2y)

     = x + 2         --→ eq 2

3: Put the result of the first partial derivative equal to zero

    For ∂/∂x [f(x,y)]

    6x + y = 0

     y = -6x

    For ∂/∂y [f(x,y)]

     x + 2 = 0

     x = -2

Put the value of x in (i) 

     y = -6(-2) = 12 

Hence the value of roots are:  x = -2, y = 12

Example 3: 9x^2 + 8xy + 8y 

1: Compute first derivative w.r.t x: 

     = ∂/∂x (9x^2 + 8xy + 8y)

     = 18x + 8y         --→ eq 1 

2: Now again compute derivative w.r.t y: 

     =∂/∂y (9x^2 + 8xy + 8y)

     = 8x + 8         --→ eq 2

3: Put the result of the first partial derivative equal to zero

    For ∂/∂x [f(x,y)]

    18x + 8y = 0    —---> i

    For ∂/∂y [f(x,y)]

    8x + 8 = 0

    8x = -8

     x = -1     

Put the value of x in (i) 

    18(-1) + 8y =  0 

   -18 = -8y

    y  = -18 - 8 = 94 

Hence the value of roots are:  x = -1, y = 94

Example 4: y^2+8 

1: Compute derivative w.r.t y: 

    = ∂/∂y (y^2+8) 

    = 2y  

2: Now to find critical points put the first derivative equal to zero: 

     2y = 0 

      y = 0

 Hence the critical point is  0

FAQ’s

Why is finding critical points significant? 

Recognizing critical points is fundamental for understanding function, which is crucial in fields like financial aspects, design, and physical science. 

How does the calculator find critical points for multivariable functions?

For multivariable functions, the calculator computes the partial derivatives of every variable and finds the points where all partial derivatives are zero. Those are the stationary points, which are assessed for maximum, minimum, or saddle points.

What are stationary points?

These points are where the derivative of a function equals zero. They may be minimum, maximum, or saddle points, and are crucial for analyzing the behavior of functions.

How do you find critical points? 

To determine the critical points of a function, take the first derivative, set it to zero, and solve for x. Check if any values make the first derivative undefined.