Local Extrema Finder

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

Function \(f(x,y,z)\): Variables:

Finds critical points where \(\nabla f=0\) (numerically), then classifies with the Hessian test. Type pi, e, sqrt(2), sin(x), x^2. Graph shows a contour map of the surface \(z=f(x,y)\) (or a slice at \(z=z_0\) in 3D).

Search domain (box)

x min: x max: y min: y max: Boundary check:

Solver & graph settings (optional)

Seeds per axis: Newton iterations: Gradient tolerance: Dedup radius: Graph samples: Contour levels: Show grid: Shaded heatmap:

Ready

Enter \(f\) and a domain, then click “Calculate”.

Graph

Contour map of \(z=f(x,y)\)

Drag to pan, mouse wheel to zoom, double-click to reset. Hover to read \((x,y)\) and \(f\). Critical points found are marked and labeled.

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Theory

Critical points and the gradient test

A critical point of a differentiable function \(f\) occurs where the gradient is zero:

\[ \nabla f(x,y)=\left\langle f_x, f_y\right\rangle = \mathbf{0} \qquad\text{or (3D)}\qquad \nabla f(x,y,z)=\left\langle f_x, f_y, f_z\right\rangle = \mathbf{0}. \]

At a critical point, the surface may have a local minimum, a local maximum, or a saddle point.

This calculator finds critical points numerically (multi-seed Newton method) and then classifies them using the Hessian test.

Second derivative test in 2 variables (Hessian determinant)

For \(f(x,y)\), the Hessian is:

\[ H=\begin{pmatrix} f_{xx} & f_{xy}\\ f_{xy} & f_{yy} \end{pmatrix}. \]

Define the Hessian determinant:

\[ D = f_{xx}f_{yy} - (f_{xy})^2. \]

Condition at the critical point	Classification
\(D>0\) and \(f_{xx}>0\)	Local minimum
\(D>0\) and \(f_{xx}<0\)	Local maximum
\(D<0\)	Saddle point
\(D=0\)	Inconclusive (degenerate case)

When \(D=0\), the test fails: higher-order analysis or different methods are needed.

3 variables: definiteness of the Hessian

For \(f(x,y,z)\), the Hessian is a symmetric \(3\times 3\) matrix of second partial derivatives. Classification depends on whether \(H\) is positive definite, negative definite, or indefinite:

Positive definite Hessian \(\Rightarrow\) local minimum.
Negative definite Hessian \(\Rightarrow\) local maximum.
Indefinite Hessian \(\Rightarrow\) saddle point.
If the Hessian is singular / near-singular \(\Rightarrow\) degenerate (inconclusive).

A convenient sufficient criterion for positive/negative definiteness is Sylvester’s criterion using principal minors (as implemented in the calculator).

Worked example

Let \(f(x,y)=x^2+y^2-2x-4y+5\). Then:

\[ f_x=2x-2,\qquad f_y=2y-4. \]

Solve \(\nabla f=\mathbf{0}\):

\[ 2x-2=0\Rightarrow x=1,\qquad 2y-4=0\Rightarrow y=2. \]

The Hessian is constant:

\[ f_{xx}=2,\quad f_{yy}=2,\quad f_{xy}=0,\quad D=2\cdot2-0^2=4>0,\ \ f_{xx}>0, \]

so \((1,2)\) is a local minimum. The minimum value is:

\[ f(1,2)=1+4-2-8+5=0. \]

Boundary checks (absolute extrema on closed regions)

If you want absolute maxima/minima on a closed and bounded region (like a rectangle), you must check:

Interior critical points \((\nabla f=0)\)
Boundary behavior (edges and corners)

The calculator’s boundary option performs a dense edge scan and reports the smallest/largest sampled values to help you locate candidates quickly.

Frequently Asked Questions

What is a critical point in multivariable calculus?

A critical point is a point where the gradient is zero, meaning all first partial derivatives vanish at that location. It is a candidate for a local minimum, local maximum, or saddle point.

How does the Hessian test classify local maxima and minima for f(x,y)?

For f(x,y), the test uses D = f_xx f_yy - (f_xy)^2 at the critical point. If D > 0 and f_xx > 0 it is a local minimum, if D > 0 and f_xx < 0 it is a local maximum, and if D < 0 it is a saddle point.

What does the boundary check option do in 2D?

Boundary check samples the edges of the rectangular domain to look for small or large values that may indicate absolute minima or maxima on a closed region. It complements interior critical points found from grad f = 0.

Why does the tool find critical points numerically instead of solving exactly?

Many multivariable equations from grad f = 0 do not have simple closed-form solutions. A multi-seed Newton method can locate critical points reliably within a specified domain, then the Hessian test is used for classification.