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Saddle Point Calculator

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

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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."

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.
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