Multivariable Extrema Calculator

📖 Tutorial | Multivariable Extrema

1. Theorem

Second-derivative test: if f_x=f_y=0 and partials are continuous, D=f_xx f_yy−f_xy². D>0 & f_xx>0 min; D>0 & f_xx<0 max; D<0 saddle; D=0 inconclusive.

2. Symbols

critical	f_x=f_y=0
D	Hessian discriminant
D>0	extremum
D<0	saddle

3. How it works

Check f_x,f_y≈0;
Compute three second partials;
Compute D and classify.

4. Steps

Enter f(x,y);
Enter the candidate critical point;
Click Calculate.

5. Example

Example: f=x³+y³−3xy at (1,1).
Solution: f_xx=6, f_yy=6, f_xy=−3, D=27>0 and f_xx>0 → local minimum.

6. Pitfalls

Be sure it is a critical point first;
D=0 is inconclusive;
A saddle is neither max nor min.

❓ FAQ | Multivariable Extrema

How to find critical points?

Solve f_x=0, f_y=0.

Why look at f_xx when D>0?

f_xx and f_yy share sign; it decides min vs max.

What is a saddle?

Max in one direction, min in another.

D=0?

Inconclusive; use higher order or definition.

Auto-find critical points?

It classifies a point you supply.