What is a critical point?

A critical point of a differentiable function is a point where its gradient is zero.

Does the starting point affect the result?

Yes. Numerical Newton iteration begins at the supplied starting point. Functions with multiple critical points can produce different results from different starting points.

How does the calculator classify critical points?

The calculator evaluates the Hessian matrix at the critical point and examines its eigenvalues. Positive eigenvalues indicate a local minimum, negative eigenvalues indicate a local maximum, and mixed signs indicate a saddle point.

Critical Point Calculator | Find and Classify Critical Points

How It Works

A critical point of a differentiable function is a point where the gradient is zero:

\[ \nabla f(x)=0. \]

For a function of two variables, \[ f(x,y), \] the calculator solves \[ f_x(x,y)=0 \] and \[ f_y(x,y)=0. \] The numerical solver starts at the coordinates you enter and iteratively searches for a zero of the gradient. This means the starting point is important when the function has several critical points. Newton's method is an iterative method whose next point is computed from the current point, so the initial guess is an actual part of the numerical calculation.

Example With Three Critical Points

\[ f(x,y)=x^4-2x^2+y^2 \] \[ \nabla f= (4x^3-4x,\;2y) \] Therefore, \[ 4x^3-4x=0 \] gives \[ x=-1,\;0,\;1 \] and \[ y=0. \] Thus the critical points are \[ (-1,0),\quad(0,0),\quad(1,0). \] For this example: \[ (0.8,0.5)\rightarrow(1,0) \] \[ (-0.8,0.5)\rightarrow(-1,0) \] \[ (0.1,0.5)\rightarrow(0,0). \]

Hessian Classification

After finding a critical point, the calculator evaluates the Hessian matrix. Its eigenvalues are used for the standard second-derivative test.

All eigenvalues positive → local minimum.
All eigenvalues negative → local maximum.
Positive and negative eigenvalues → saddle point.
An eigenvalue close to zero → degenerate or inconclusive case.

Related Multivariable Calculators

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Frequently Asked Questions

Why does the starting point matter?

Newton's method is an iterative numerical method. It begins with the initial point supplied by the user and calculates successive approximations. Therefore, when several critical points exist, different initial points can converge to different critical points.

Will my starting point be preserved after refresh?

Yes. This calculator uses a GET request for the calculation. The function and starting point are therefore stored in the URL and the form is populated directly from those submitted values.

Can I enter several starting points?

Yes. Separate them with semicolons. For example:


0.8,0.5; -0.8,0.5; 0.1,0.5

Why can two starting points reach the same critical point?

That is normal. A numerical solver can have a basin of attraction around a particular solution. Different starting points inside that basin can converge to the same critical point.