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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:**
**Output:**
The saddle point calculator calculates:
## 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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- 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

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