# Critical Point Calculator

Input the function and click calculate button to find the critical points using critical point calculator

This will be calculated:

Critical point calculator is used to find the critical points of one or multivariable functions at which the function is not differentiable. This critical point calculator gives the step-by-step solution along with the graph.

## What is the critical point?

In calculus, a critical point of a continuous function is a point at which the derivative is zero or undefined. On the graph, the critical points are the points where the rate of change of function is altered.

## How to calculate a critical point?

Below are a few solved examples of the critical point.

### Example 1: For one variable function

Find the critical point of x^2+2x+4.

Solution

Step 1: Take the derivative of the given one-variable function.

d/dx[x^2+2x+4] = d/dx (x^2) + d/dx (2x) + d/dx (4)
d/dx[x^2+2x+4] = 2x + 2 + 0
d/dx[x^2+2x+4] = 2x + 2

Step 2: Find the critical point by putting d/dx[f(x)] = 0

d/dx[x^2+2x+4] = 0
2x + 2 = 0
2x = -2
x = -2/2 = -1

Hence, x = -1 is the critical point of the given one-variable function.

### Example 2: For two-variable functions

Find the critical point of 3x^2+2xy+6y.

Solution

Step 1: Take the partial derivative of the given function with respect to x.

∂/∂x [4x^2+2xy+6y] = ∂/∂x (4x^2) + ∂/∂x (2xy) + ∂/∂x (6y)
∂/∂x [4x^2+2xy+6y] = 8x + 2y + 0
∂/∂x [4x^2+2xy+6y] = 8x + 2y

Step 2: Take the partial derivative of the given function with respect to y.

∂/∂y [4x^2+2xy+6y] = ∂/∂y (4x^2) + ∂/∂y (2xy) + ∂/∂y (6y)
∂/∂y [4x^2+2xy+6y] = 0 + 2x + 6
∂/∂y [4x^2+2xy+6y] = 2x + 6

Step 3: Find the critical points by putting f’(x, y) = 0

∂/∂x [f(x, y)] = 2x + 6 = 0
2x + 6 = 0
2x = -6
x = -6/2 = -3
x = -3

∂/∂y [f(x, y)] = 8x + 2y = 0
8x + 2y = 0
8(-3) + 2y = 0
-24 + 2y = 0
2y = 24
y = 24/2
y = 12

Hence, x = -3 & y = 12 are the critical points of the given two-variable function. 