Critical Number Calculator

Find critical points of \(f(x,y)\) by solving \(\nabla f=(0,0)\) and classify them via the Hessian test.

Function \(f(x,y)\)

Use standard math: x^2, sin(x), exp(y), x*y, etc. Trig uses radians.

Helping notes

Critical points satisfy \(f_x=0\) and \(f_y=0\).

Hessian test: \(D=f_{xx}f_{yy}-f_{xy}^2\).

If \(D>0\) and \(f_{xx}>0\) ⇒ local min; if \(D>0\) and \(f_{xx}<0\) ⇒ local max.

If \(D<0\) ⇒ saddle; if \(D=0\) ⇒ inconclusive.

Results

Critical Points Found

Gradient & Hessian (at each point)

Classification

Equation Preview

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What is a Critical Number Calculator?

A Critical Number Calculator finds the x-values where a function’s behavior can change dramatically—possible local maxima, minima, or nondifferentiable corners. For a single-variable function \(f(x)\), a critical number \(c\) is any domain point where the derivative is zero, \(f'(c)=0\), or the derivative does not exist while \(f\) remains defined at \(c\). These locations are candidates for local extrema by Fermat’s Theorem, which states that if \(f\) has a local extremum at an interior point and is differentiable there, then the derivative must be zero. The calculator automates differentiation, solves the necessary equations/conditions, and then helps you classify the points using second-derivative and first-derivative sign tests. It also distinguishes between interior critical numbers and endpoints, which are not critical numbers but are still candidates in closed-interval optimization.

About the Critical Number Calculator

The tool accepts symbolic functions (polynomials, rational, exponential, logarithmic, trigonometric) and handles piecewise definitions with domain checks. It computes \(f'(x)\), solves \(f'(x)=0\), and flags points where \(f'\) fails to exist while \(f\) is defined. For classification, it evaluates \(f''(x)\) and applies the second-derivative test; when inconclusive, it performs a first-derivative sign analysis around the candidate. On closed intervals \([a,b]\), it compiles the standard comparison set: all critical numbers in \((a,b)\) plus the endpoints \(a,b\), then reports absolute maxima/minima by direct evaluation. The output includes clear algebraic steps, domain warnings (e.g., vertical asymptotes), and a concise summary table of each candidate’s type and function value.

How to Use this Critical Number Calculator

Enter your function \(f(x)\) and (optionally) an interval \([a,b]\) for absolute extrema checks.
Compute \(f'(x)\) and solve \(f'(x)=0\); the tool lists all real solutions in the domain of \(f\).
Include nondifferentiable points where \(f\) is defined (cusps, corners) as additional candidates.
Classify candidates: use \(f''(c)\) or a one-sided sign test on \(f'(x)\) across \(c\).
If optimizing on \([a,b]\), also evaluate \(f(a)\) and \(f(b)\) and compare all values.

Core Formulas (LaTeX)

Critical numbers (interior): \[ \mathcal{C}=\Big\{\,x\in\mathrm{dom}(f):\ f'(x)=0\ \ \text{or}\ \ f'(x)\ \text{does not exist}\,\Big\}. \]

Second-derivative test: \[ f''(c)>0\Rightarrow\text{local min},\qquad f''(c)<0\Rightarrow\text{local max},\qquad f''(c)=0\Rightarrow\text{inconclusive}. \]

First-derivative test: \[ \begin{cases} f'(x)\ \text{changes }(-\to +)\ \text{at }c \Rightarrow \text{local min}\\ f'(x)\ \text{changes }(+\to -)\ \text{at }c \Rightarrow \text{local max}\\ \text{no sign change} \Rightarrow \text{neither} \end{cases} \]

Closed-interval comparison set: \[ \{\text{critical numbers in }(a,b)\}\ \cup\ \{a,b\}. \]

Examples (Illustrative)

Example 1 — Polynomial

\(f(x)=x^3-3x\). \(f'(x)=3x^2-3=0\Rightarrow x=\pm1\). \(f''(x)=6x\). \(f''(-1)=-6<0\Rightarrow\) local max at \(-1\); \(f''(1)=6>0\Rightarrow\) local min at \(1\).

Example 2 — Nondifferentiable point

\(f(x)=|x|\). \(f'(x)=-1\) for \(x<0\), \(f'(x)=1\) for \(x>0\); derivative fails at \(x=0\) while \(f\) is defined. Thus \(0\) is a critical number (and a global minimum).