fdFeedback
In wa

 

Adblocker Detected

 

ad

Uh Oh! It seems you’re using an Ad blocker!

Since we’ve struggled a lot to makes online calculations for you, we are appealing to you to grant us by disabling the Ad blocker for this domain.

derivative Calculator

Saddle Point Calculator

Enter a function f(x,y):

keyboard

Table of Content

Get the Widget!

Add this calculator to your site and lets users to perform easy calculations.

Feedback

How easy was it to use our calculator? Did you face any problem, tell us!

An online saddle point calculator helps you to determine the saddle point of a multivariable function.

Let us focus on the conceptual structure of the saddle point.

What Is a Saddle Point In Calculus?

In the light of saddle point calculus,

a point where the second partial derivatives of a multivariable function become zero with no minimum or maximum value.”

Saddle Point Equation:

You can find saddle point when the following condition is fulfilled:

$$ \frac{\partial^{2}}{\partial {(x,y)}^{2}}\ F{\left(x, y\right)} = 0 $$

How To Find a Saddle Point?

Finding saddle points is somehow or what a little bit tricky but not tough. Let us solve the following saddle point

example to get a hands-on grip.

Example # 01:

Find saddle point for the function given below:

$$ F{\left(x, y\right)} = \left(x^{3} – 5 x y^{2} + y\right) $$

Solution:

As we already know that the condition for a saddle point is:

$$ \frac{\partial^{2}}{\partial {(x,y)}^{2}} F{\left(x, y\right)} = 0 $$

For the function given, we have:

$$ \frac{\partial^{2}}{\partial {(x,y)}^{2}} \left(x^{3} – 5 x y^{2} + y\right) = 0 $$

1st derivative steps w.r.t x:

$$ \frac{\partial}{\partial x}\left(x^{3} – 5 x y^{2} + y\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial x}\left(x^{3} – 5 x y^{2} + y\right) = 3x^2 + 5y^2 $$

2nd derivative w.r.t.x:

$$ \frac{\partial}{\partial x}\left(3 x^{2} – 5 y^{2}\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial x}\left(3 x^{2} – 5 y^{2}\right) = 6x $$

now, we have;

1st partial derivative w.r.t y:

$$ \frac{\partial}{\partial y}\left(x^{3} – 5 x y^{2} + y\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial y}\left(x^{3} – 5 x y^{2} + y\right) = -10x + 1 $$

2nd partial derivative w.r.t y:

$$ \frac{\partial}{\partial y}\left(- 10 x y + 1\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial y}\left(- 10 x y + 1\right) = -10x $$

Finding saddle points:

To find saddle points put f”(x,y) = 0

6x=0

x = 0 / 6

x = 0

−10x=0

x = 0 / -10

x = 0

Roots: {x:0}

Which is the required saddle point.

If you are looking for instant results, use online saddle point calculator.

Example # 02:

Find saddle points of the function below:

$$ F{\left(x, y\right)} = \left(x^{4} – 5 x y + y^{3}\right) $$

Solution:

We know that for saddle points:

$$ \frac{\partial^{2}}{\partial {(x,y)}^{2}} F{\left(x, y\right)} = 0 $$

For the function given, we have:

$$ \frac{\partial^{2}}{\partial {(x,y)}^{2}} \left(x^{4} – 5 x y + y^{3}\right) = 0 $$

1st derivative steps w.r.t x:

$$ \frac{\partial}{\partial x}\left(x^{4} – 5 x y + y^{3}\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial x}\left(x^{4} – 5 x y + y^{3}\right) = 4x^{3} – 5y $$

2nd derivative w.r.t.x:

$$ \frac{\partial}{\partial x}\left(4 x^{3} – 5 y\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial x}\left(4 x^{3} – 5 y\right) = 12x^{2} $$

1st partial derivative w.r.t y:

$$ \frac{\partial}{\partial y}\left(x^{4} – 5 x y + y^{3}\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial y}\left(x^{4} – 5 x y + y^{3}\right) = -5x + 3y^{2} $$

2nd partial derivative w.r.t y:

$$ \frac{\partial}{\partial y}\left(- 5 x + 3 y^{2}\right) $$ (click partial derivative calculator for calculations)

The derivative is:

$$ \frac{\partial}{\partial y}\left(- 5 x + 3 y^{2}\right) = 6y $$

Finding saddle points:

To find saddle points put f”(x,y) = 0

12x^{2}=0

x = 0

6y = 0

y = 0

Roots: {x:0, y:0}

If you have any doubt about the calculations you performed, you can verify the results using our free online saddle point calculator.

How a Saddle Point Calculator Works?

Performing manual calculations to find saddle points may take a lot of time. Apart from this, we have introduced you to a free online saddle points calculator.

Let us see what we need to do:

Input:

  • Write your functions in the menu bar
  • Click’ calculate’

Output:

The saddle point calculator calculates:

  • First-order derivatrive w.r.t x
  • Second-order derivative w.r.t x
  • First-order derivative w.r.t y
  • Second-order derivative w.r.t y
  • Step by step calculations
  • Saddle point for the function given

FAQ’s:

What is a saddle point example in real life?

In the real-world, the surface of a handkerchief is a good example of a saddle point.

What do you mean by extremum?

The point where we can get the minimum or maximum value of a function is termed as extremum.

How do you classify extremum?

For each value, you have to test an x-value slightly smaller and slightly larger than that x-value. If both are smaller than f(x), then it is a maximum. If both are larger than f(x), then it is a minimum.

Is every turning point a saddle point?

There are two types of stationary points: saddle points and turning points. While turning points correspond to local extrema, saddle points do not

Conclusion:

In saddle points calculus, a saddle point or minimax point is a point on the surface of the graph for a function where the slopes in perpendicular directions become zero (acritical point), but which is not a local extremum of the function. Mathematicians and engineers always have to find saddle point when doing an analysis of a surface. For this purpose, our free online saddle points calculator is a beneficial tool designed so far.

References:

From the source of Wikipedia: Saddle surface, Maxima and minima, Functions of more than one variable.

From the source of khan academy: Second partial derivative test, Loose intuition, Gradient descent

From the source of lumen learning: Functions of Several Variables, Limits and Continuity, Partial Derivatives, Linear Approximation, The Chain Rule, Maximum and Minimum Values, Lagrange Multiplers, Optimization in Several Variables.