Critical Point Finder

Math Calculus • Applications of Derivatives

Written by STEM Calculators Team Published December 23, 2025 Updated February 24, 2026

1. Critical Point Finder

Finds critical points by solving \(f'(x)=0\) (and detecting non-differentiable points numerically), then classifies via \(f''(x)\) and a first-derivative sign chart.

Inputs

Function \(f(x)\)

Supported: + − * / ^, parentheses, variable x, constants pi, e, sin cos tan, ln log(base 10), sqrt abs exp. Implicit multiplication: 2x, (x+1)(x-1), 2sin(x). Trig powers like cos^2(2x) are supported.

Output format

LaTeX is best for readability.

Classification

If \(f''(x_0)=0\), sign test is often needed.

Search interval Restrict to \([a,b]\)

If unchecked, uses the plot window \([c-w,c+w]\).

\(a\)

\(b\)

Endpoints Consider endpoints for absolute max/min

Endpoints are not “critical points” but can be extrema on \([a,b]\).

Plot center \(c\)

Plot half-width \(w\)

Plots \(x\in[c-w,c+w]\).

Graph options Show grid Show \(f'(x)\) graph Label critical points

Quick presets

Click to auto-fill and compute.

Random function

Picks a sample function and interval.

Ready

Graphs

Drag to pan • wheel/pinch to zoom • wide zoom-out enabled. Left: \(f(x)\). Right: \(f'(x)\) (optional). X-view is shared.

\(f(x)\)

\(f'(x)\)

x: 0, y: 0, zoom(px/unit): 45

Result

Enter \(f(x)\) and click Find critical points.

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1. Critical Point Finder — Theory

Critical points help locate local maxima/minima and analyze increasing/decreasing behavior. This page summarizes the tests used by the calculator.

1) Critical points

A number \(x=c\) is a critical point of \(f\) (in the usual single-variable sense) if:

\(f(c)\) exists, and
either \(f'(c)=0\) or \(f'(c)\) does not exist (but \(f(c)\) still exists).

These are the interior “candidates” for local extrema. On a closed interval \([a,b]\), you must also check endpoints \(a\) and \(b\) for absolute extrema.

2) How to find critical points

For differentiable parts of \(f\), solve the equation:

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

If \(f\) has corners/cusps/vertical tangents (common with \(|x|\), piecewise behavior, radicals, etc.), you also look for points where \(f'(x)\) is undefined while \(f(x)\) exists.

3) Second-derivative test

Suppose \(f'(c)=0\) and \(f''(c)\) exists. Then:

\[ f''(c) > 0 \Rightarrow \text{local minimum at }c, \qquad f''(c) < 0 \Rightarrow \text{local maximum at }c. \]

If \(f''(c)=0\), the test is inconclusive. You then use the first-derivative sign test or higher-order reasoning.

4) First-derivative sign test

Look at the sign of \(f'(x)\) just to the left and right of \(c\):

\(f'\) changes from \(+\) to \(-\): local maximum at \(c\).
\(f'\) changes from \(-\) to \(+\): local minimum at \(c\).
No sign change: \(c\) is not a local extremum (often a flat/saddle-type point).

A sign chart partitions the interval using critical points and checks the sign of \(f'\) on each sub-interval. This also tells you where \(f\) is increasing (\(f'>0\)) or decreasing (\(f'<0\)).

5) Multiplicity intuition (why some roots don’t switch sign)

If \(f'(x)\) has a root of even multiplicity, the derivative may touch zero and turn around without changing sign. Then \(f\) can remain increasing on both sides (or decreasing on both sides), producing a flat/saddle-type critical point.

If the root is odd multiplicity, \(f'(x)\) typically crosses the axis and changes sign, producing a local max/min.

6) Absolute extrema on a closed interval

On \([a,b]\), absolute maxima/minima occur among:

critical points in \((a,b)\), and
endpoints \(a\) and \(b\).

Compute \(f\) at each candidate and compare values.

7) Worked example

Let \(f(x)=x^3-3x\). Then:

\[ f'(x)=3x^2-3=3(x-1)(x+1)\Rightarrow f'(x)=0 \text{ at } x=\pm 1. \]

Second derivative:

\[ f''(x)=6x,\quad f''(1)=6>0 \Rightarrow \text{local min at }x=1,\quad f''(-1)=-6<0 \Rightarrow \text{local max at }x=-1. \]

Frequently Asked Questions

What is a critical point in calculus?

A number c is a critical point of f if f(c) exists and either f'(c)=0 or f'(c) does not exist. Critical points are key candidates for local maxima and minima.

How does the critical point finder solve f'(x)=0?

The calculator differentiates the input function to get f'(x) and then numerically searches for roots of f'(x)=0 in the selected window or restricted interval. It reports the x-values where the derivative is zero as critical-point candidates.

Why can a critical point occur where the derivative is undefined?

If f(x) is defined at c but has a corner, cusp, or vertical tangent, then f'(c) may not exist even though f(c) exists. Those locations are still critical-point candidates because the function can change direction there.

How do the first-derivative and second-derivative tests differ?

The second-derivative test uses the sign of f''(c) when f'(c)=0: f''(c)>0 suggests a local minimum and f''(c)<0 suggests a local maximum. The first-derivative sign test checks whether f'(x) changes sign across c, which also works when f''(c)=0 or is inconclusive.

Do endpoints count as critical points for absolute maxima or minima?

Endpoints are not critical points in the interior sense, but on a closed interval [a,b] the absolute maximum or minimum can occur at a or b. Enabling the endpoint option includes them as candidates for absolute extrema comparisons.

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