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derivative Calculator

Critical Point Calculator

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An online critical point calculator helps you to determine the local minima, maxima, stationary and critical points of the given function. This critical point finder differentiates and applies the power rule for determining the different points. .  With this guide you people come to know how to find critical points of a function using derivative and power rule and much more!

What are Critical Points?

The critical point is a wide term used in many fields of mathematics. When it comes to functions of real variables, the critical point is the point in the function domain where the function is not differentiable. When dealing with complex variables, the critical point is also the point where the function domain is not holomorphic or its derivative is zero.

Similarly, for a function of multiple real variables, the critical point is the critical value within its range (where the gradient is undefined or equal to zero). The critical point of a multidimensional function is the point where the first-order partial derivative of the function is zero.

Critical Points Classification:

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The Critical Point of the Function of a Single Variable:

The critical point of the function of a single real variable f(x) is the value of x in the region of f, which is not differentiable, or its derivative is 0 (f’ (X) = 0).

Example:

Find the critical numbers of the function 4x^2 + 8x.

Solution

Finding critical points for 4x^2 + 8x

Derivative Steps of:

$$ ∂/∂x (4x^2 + 8x) $$

Critical point calculator takes Derivative of 4x^2 + 8x term by term:

So, the derivative of a constant function is the constant times the derivative of the function.

Now, critical number calculator applies the power rule: x^2 goes to 2x

So, the result is: 8x

Then critical points calculator applies the power rule: x goes to 1

Hence, the x is: 8

The result is: 8x + 8

Finally, critical numbers calculator finds critical points by putting f'(x) = 0

8x + 8 = 0

Local Minima

(x, f(x)) = (−1, −4.0)

Local Maxima

(x, f(x)) = No local maxima

Roots: [−1]

How to Calculate the Critical Points for Two Variables?

To find these points manually you need to follow these guidelines:

  • First, write down the given function and take the derivative of all given variables.
  • Now, apply the power rule after differentiation.
  • Then, finds the local minima and maxima by substituting 0 in the place of variables.

However, you can find these points with our critical points calculator by following these steps:

Example:

 Find the critical points for multivariable function: 4x^2 + 8xy + 2y.

Solution:

Derivative Steps of:

∂/∂x (4x^2 + 8xy + 2y)

Multivariable critical point calculator differentiates 4x^2 + 8xy + 2y term by term:

The critical points calculator applies the power rule: x^2 goes to 2x

So, the derivative is: 8x

Again, the critical number calculator applies the power rule: x goes to 1

The derivative of 8xy is: 8y

The derivative of the constant 2y is zero.

So, the result is: 8x + 8y

Now, the critical numbers calculator takes the derivative of the second variable:

∂/∂y (4x^2 + 8xy + 2y)

Differentiate 4x^2 + 8xy + 2y term-by-term:

The derivative of the constant 4x^2 is zero.

Now, apply the power rule: y goes to 1

So, the derivative is: 8x

Apply the power rule: y goes to 1

Hence, the derivative of 2y is: 2

The answer is: 8 x + 2

To find critical points put f'(x, y) = 0

8x + 8y = 0

8x + 2 = 0

So, the critical numbers of a function are:

Roots: {x:−14, y:14}

How Critical Points Calculator Works?

An online critical number calculator finds the critical points with several methods by following these guidelines:

Input:

  • First, enter any function with single or multiple variables.
  • Click on the calculate button to see the step-wise calculations.

Output:

  • The critical point calculator displays the critical points for the given function.
  • It uses the derivative and power rule for determining the critical and stationary points.

FAQ:

What are the types of critical points?

Critical points are places where ∇f or ∇f=0 does not exist. The critical point is the tangent plane of points z = f(x, y) is horizontal or does not exist. All local extrema and minima are the critical points.

  • Local minima at (−π2,π2),(π2,−π2),
  • Local maxima at (π2,π2),(−π2,−π2),
  • A saddle point at (0,0).

What if there is no critical point?

If the function has no critical point, then it means that the slope will not change from positive to negative, and vice versa. So, the critical points on a graph increases or decrease, which can be found by differentiation and substituting the x value.

Conclusion:

Use this online critical point calculator that provides critical points for both single and multiple variable functions. It uses different methods for determining the local maxima and minima for a given single variable function precisely.

Reference:

From the source of Wikipedia: Critical point of a single variable function, Location of critical points, Critical points of an implicit curve, Use of the discriminant.

From the source of Brilliant: point of a continuous function, optimization problems, differentiable function f, local extremum, an inflection point.