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Write down any multivariable function and the calculator will find its saddle point, with calculations displayed.

Enter a function f(x,y):

Table of Content

 1 What Is a Saddle Point In Calculus? 2 Saddle Point Equation: 3 How To Find a Saddle Point? 4 What is a saddle point example in real life? 5 What do you mean by extremum? 6 How do you classify extremum? 7 Is every turning point a saddle point?

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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.”

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:

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

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:

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

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:

• Click’ calculate’

Output:

• 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.