Critical Point Calculator

Critical points calculator with steps

Critical points calculator finds the values of single or multivariable functions. This critical number calculator determines those points on which the function is not differentiable. 

What is the critical point?

A critical point is the point of the function at which the differential of the function is zero or undefined. It can also be defined as a point on the graph of a function where the differentiation is zero or infinite. 

Critical points can be calculated by putting the first derivative equal to zero.

f'(x) = 0

How to calculate the critical point?

To learn how to calculate the critical points, follow the below examples.

Example 1

Calculate the critical point of 3x^2 + 4x + 9.

Solution 

Step I: First of all, find the first derivative of the given function.

d/dx [3x^2 + 4x + 9] = d/dx [3x^2] + d/dx [4x] + d/dx [9]

d/dx [3x^2 + 4x + 9] = 6x + 4 + 0

d/dx [3x^2 + 4x + 9] = 6x + 4

Step II: Now calculate the critical point by substituting the first derivative equal to zero.

d/dx [3x^2 + 4x + 9] = 0

6x + 4 = 0

6x = -4

x = -4/6 = -2/3

Hence, the critical point of the given function is x = -2/3

Example 2:

Calculate the critical point of 4x^2 + 6xy + 8y.

Solution 

Step I: First of all, calculate the first partial derivative of the function w.r.t “x”.

∂/∂x [4x^2 + 6xy + 8y] = ∂/∂x [4x^2] + ∂/∂x [6xy] + ∂/∂x [8y]

∂/∂x [4x^2 + 6xy + 8y] = 8x + 6y + 0

∂/∂x [4x^2 + 6xy + 8y] = 8x + 6y

Step II: Now calculate the first partial derivative of the function w.r.t “y”.

∂/∂y [4x^2 + 6xy + 8y] = ∂/∂y [4x^2] + ∂/∂y [6xy] + ∂/∂y [8y]

∂/∂y [4x^2 + 6xy + 8y] = 0 + 6x + 8

∂/∂y [4x^2 + 6xy + 8y] = 6x + 8

Step III: Put the result of the first partial derivatives equal to zero.

For ∂/∂x [f(x, y)]

8x + 6y = 0

6y = -8x

y = -8x/6 … (i)

For ∂/∂y [f(x, y)]

6x + 8 = 0

6x = -8

x = -8/6 = -4/3

put the value of “x” in (i)

y = -8(-4/3)/6

y = 32/18

y = 16/9

Hence the critical points of the given function are:

x = -4/3

y = 16/9