Critical Point Calculator

Critical Point Calculator

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.