Saddle Point Calculator

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 −10

x=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.