Example 1 — Univariate cubic
\(f(x)=x^3-3x\). \(f'(x)=3x^2-3=0\Rightarrow x=\pm1\). \(f''(x)=6x\). At \(x=-1\): \(f''=-6<0\Rightarrow\) local max, \(f(-1)=2\). At \(x=1\): \(f''=6>0\Rightarrow\) local min, \(f(1)=-2\).
Critical Points Calculator
Enter f(x,y). We solve ∇f = (fx, fy) = (0,0) with a multi-start Newton method and classify using the Hessian:
D = fxxfyy − (fxy)².
Use math like
x^2, y^2, sin(x*y), exp(x), log(y).Left bound for initial guesses.
Right bound for initial guesses.
Bottom bound for initial guesses.
Top bound for initial guesses.
Initial grid density (x × y).
Newton step limit.
Stop when |∇f| and step size are below this.
Display precision only.
Helping Notes
- If nothing is found, widen the search box or increase seeds.
- Non-smooth terms (e.g.,
abs) can make derivatives undefined. - Very flat regions may be “inconclusive” (near-zero Hessian determinant).
Result
Equation Preview
We symbolically compute
fx, fy, fxx, fxy, fyy.
Critical points satisfy fx=0 and fy=0.
Critical Points
Classification via Hessian: if
D>0 and fxx>0 → local minimum;
if D>0 and fxx<0 → local maximum; if D<0 → saddle.
Values & Checks
Residuals show
‖∇f(x*,y*)‖. Smaller is better.
What is a Critical Points Calculator?
A Critical Points Calculator locates and classifies the special points of a function where behavior can change—local maxima, local minima, or saddle points. In one variable, critical points occur where the derivative is zero or undefined (within the domain). In several variables, they occur where the gradient is the zero vector (or where partial derivatives fail to exist while the point remains in the domain). The calculator computes first derivatives, solves for candidate points, and then applies second-derivative or Hessian tests to label each candidate as a max, min, saddle, or inconclusive..
About the Critical Points Calculator
For \(f:\mathbb{R}\to\mathbb{R}\), critical points satisfy \(f'(x)=0\) (or derivative undefined). Classification uses the second-derivative test: \(f''(x^\*)\gt 0\) indicates a local minimum; \(f''(x^\*)\lt 0\) a local maximum; \(f''(x^\*)=0\) is inconclusive. For \(f:\mathbb{R}^n\to\mathbb{R}\), candidates solve \(\nabla f(\mathbf{x})=\mathbf{0}\). The Hessian matrix \(H(\mathbf{x})\) determines type via definiteness: positive definite \(\Rightarrow\) local minimum; negative definite \(\Rightarrow\) local maximum; indefinite \(\Rightarrow\) saddle; singular/inconclusive requires higher-order analysis or direct testing. The tool can also evaluate function values, check constraints (optional), and export a summary table of points and classifications.
How to Use this Critical Points Calculator
- Enter the function \(f(x)\) (1D) or \(f(x_1,\dots,x_n)\) (multi-variable). Use standard syntax for powers, roots, and trig.
- Choose Find Critical Points to solve \(\nabla f=\mathbf{0}\) (or \(f'=0\)) and list candidates.
- Select Classify to compute \(f''\) (1D) or the Hessian \(H\) (multi-D) and apply tests automatically.
- (Optional) Provide numeric ranges or initial guesses to guide solvers for complicated systems.
- Review coordinates, function values, and labels (max/min/saddle/inconclusive). Export LaTeX steps if needed.
Core Formulas
1D critical points: \[ \mathcal{C}=\{x\in\mathrm{dom}(f): f'(x)=0 \ \text{or}\ f'(x)\ \text{DNE}\}. \]
1D classification: \[ f''(x^\*)>0 \Rightarrow \text{local min},\quad f''(x^\*)<0 \Rightarrow \text{local max},\quad f''(x^\*)=0 \Rightarrow \text{inconclusive}. \]
Gradient (multi-D): \[ \nabla f(\mathbf{x})=\left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right)^{\!\top},\quad \nabla f(\mathbf{x}^\*)=\mathbf{0}. \]
Hessian and 2D test: \[ H=\big[\partial^2 f/\partial x_i\partial x_j\big],\quad D=f_{xx}f_{yy}-f_{xy}^2. \] \[ D>0,\ f_{xx}>0\Rightarrow \text{min};\quad D>0,\ f_{xx}<0\Rightarrow \text{max};\quad D<0\Rightarrow \text{saddle};\quad D=0\Rightarrow \text{inconclusive}. \]
Examples (Illustrative)
Example 2 — Positive-definite quadratic (2D)
\(f(x,y)=x^2+xy+y^2\). \(\nabla f=(2x+y,\ x+2y)=\mathbf{0}\Rightarrow (0,0)\). \(H=\begin{bmatrix}2&1\\1&2\end{bmatrix}\) is positive definite (eigenvalues \(1,3\gt0\)) \(\Rightarrow\) strict local minimum at \((0,0)\).
Example 3 — Saddle
\(f(x,y)=x^2-y^2\). \(\nabla f=(2x,-2y)=\mathbf{0}\Rightarrow (0,0)\). \(H=\mathrm{diag}(2,-2)\) is indefinite \(\Rightarrow\) saddle at \((0,0)\).
FAQs
What are critical points?
Points where \(f'(x)=0\) or undefined (1D), or \(\nabla f=\mathbf{0}\) (multi-D), indicating potential extrema or saddle behavior.
How do I classify critical points?
Use \(f''\) in 1D; use Hessian definiteness (or \(D\) in 2D) in multivariable settings.
Can there be multiple critical points?
Yes—polynomials and nonlinear systems often have several; the calculator lists and classifies each one.
What if the test is inconclusive?
Try higher derivatives, evaluate nearby values, or use eigenvalue analysis/series expansion.
Does a critical point guarantee a max or min?
No. Indefinite Hessian implies a saddle; singular cases may need deeper analysis.
Can I restrict the search region?
Yes—provide bounds or initial guesses to focus numerical solvers in complex landscapes.
Will this handle nondifferentiable corners?
It flags derivative failures and can still test one-sided behavior or compare values on piecewise domains.
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