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Critical Points Calculator

Math

Example functions

Symbolic Derivatives

f'(x)

3 * x ^ 2 - 6 * x

f''(x)

6 * x - 6

Critical Points

xf(x)ClassificationReasoning
02Local Max

f''(x) = -6.000000 < 0, so the graph is concave down here (second derivative test).

2-2Local Min

f''(x) = 6.000000 > 0, so the graph is concave up here (second derivative test).

Absolute Extrema on [-10, 10]

Absolute Maximum

702

at x = 10

Absolute Minimum

-1298

at x = -10

Step-by-Step Solution

1f(x) = x^3 - 3*x^2 + 2
2Differentiate: f'(x) = 3 * x ^ 2 - 6 * x
3Differentiate again: f''(x) = 6 * x - 6
4Solve f'(x) = 0 for x over [-10, 10]
5x = 0: f(x) = 2 → Local Maximum (f''(x) = -6.000000 < 0, so the graph is concave down here (second derivative test).)
6x = 2: f(x) = -2 → Local Minimum (f''(x) = 6.000000 > 0, so the graph is concave up here (second derivative test).)
7Absolute max on [-10, 10] = 702 at x = 10
8Absolute min on [-10, 10] = -1298 at x = -10

Graph of f(x) with Critical Points

Local Max

Local Min

Saddle

Corner

Sign of f'(x) across [-10, 10]

x = -10

f'(x) > 0 (green) · f'(x) < 0 (red)

x = 10

About This Tool

Critical Points Calculator – Locate and Classify Local Extrema

A critical point of a function f(x) is any x-value in its domain where the slope is flat or breaks — that is, where f'(x) = 0 or f'(x) is undefined. These are the only places a curve can turn from rising to falling (a local maximum), from falling to rising (a local minimum), or momentarily flatten without truly turning at all (a saddle point). This calculator parses your expression with the same symbolic engine used by the Derivative Calculator, computes f'(x) and f''(x), searches your chosen interval for every critical point, and classifies each one automatically.

How the Search Works

For simple polynomials the roots of f'(x) = 0 can sometimes be read off directly, but most real expressions — trig, exponential, logarithmic, or mixed — have no tidy closed form. Instead, the calculator densely samples f'(x) across your search interval [a, b], watches for every sign change, and refines each one with bisection down to your chosen tolerance (as small as 1e-12). It also watches for points where f'(x) is undefined even though f(x) itself is perfectly defined — the sharp corner of |x| at x = 0 is the classic example, since the left-hand and right-hand slopes there are −1 and +1 with nothing in between.

First Derivative Test vs. Second Derivative Test

The second derivative test checks the concavity at each candidate point: if f''(x₀) > 0 the curve bends upward (a local minimum), and if f''(x₀) < 0 it bends downward (a local maximum). When f''(x₀) = 0 the test is inconclusive, so the calculator automatically falls back to the first derivative test, which simply compares the sign of f'(x) just to the left and right of the point. A negative-to-positive change means a minimum, positive-to-negative means a maximum, and no change at all means the point is a saddle point — a flat spot that never actually turns, like f(x) = x³ at x = 0.

Why isn't every zero of f'(x) an extremum?
A derivative of zero only means the tangent line is momentarily horizontal. Whether that flat spot is a peak, a valley, or neither depends entirely on whether the slope actually changes sign around it — which is exactly what the first derivative test checks.

Absolute Extrema on a Closed Interval

Local extrema only describe behavior near a point, but many optimization problems ask for the single highest or lowest value f(x) reaches across a bounded range. With Include endpoints enabled, the calculator evaluates f(x) at every interior critical point plus both endpoints a and b, then reports the largest value as the absolute maximum and the smallest as the absolute minimum — the standard closed-interval method taught in every calculus course.

Concavity and Inflection Points

Turning on Show inflection points extends the same root-finding routine to f''(x) = 0, keeping only the roots where concavity genuinely flips from up to down or down to up. This complements the critical-point analysis: while critical points describe where a curve stops rising or falling, inflection points describe where it stops curving one way and starts curving the other, which is essential for accurate curve sketching.

Reading the Graph and Sign Chart

The plot draws f(x) across your search interval with each critical point marked and color-coded by type — local maxima, local minima, saddle points, and corners are all visually distinct. Beneath it, a sign chart shows where f'(x) is positive or negative across the same interval, making it easy to visually confirm exactly why the first derivative test reached the conclusion it did at each marked point.

Frequently Asked Questions

Is the Critical Points Calculator free?

Yes, Critical Points Calculator is totally free :)

Can I use the Critical Points Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Critical Points Calculator?

Yes, any data related to Critical Points Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this Critical Points Calculator work?

Enter f(x), and the tool uses the mathjs symbolic engine to compute f'(x) and f''(x), scans your search interval for every point where f'(x) equals zero or is undefined, and classifies each one with the first and/or second derivative test as a local maximum, local minimum, saddle point, or corner.

What is a critical point, exactly?

A critical point is an x-value in the domain of f where the derivative f'(x) is either zero (the tangent line is horizontal) or undefined (a sharp corner or cusp). These are the only places a local maximum or local minimum can occur.

Why do some critical points show up as 'saddle point' instead of max or min?

A saddle (non-extremum) point happens when f'(x) = 0 but the sign of f'(x) does not change on either side of it — for example f(x) = x³ at x = 0. The slope flattens momentarily but the function keeps increasing (or decreasing) through the point, so it is neither a peak nor a valley.

What is the difference between the first and second derivative test?

The second derivative test checks the sign of f''(x0): positive means concave up (local min), negative means concave down (local max), and zero is inconclusive. The first derivative test instead checks whether the sign of f'(x) changes from negative to positive (local min), positive to negative (local max), or not at all (saddle) around x0. This tool applies whichever method you choose, and automatically falls back to the first derivative test whenever the second derivative test is inconclusive.

How accurate is the root finding for functions without a simple formula?

The calculator densely samples f'(x) across your search interval, detects every sign change, and refines each one with bisection down to your chosen tolerance (default 1e-8). This numeric approach reliably finds roots for polynomials, trig, exponential, and logarithmic expressions alike, though very closely spaced roots in a narrow interval may occasionally be merged — narrow the interval or tighten the tolerance if that happens.

Does the tool handle absolute extrema on a closed interval?

Yes. When 'Include endpoints' is on, the calculator evaluates f(x) at every interior critical point plus both endpoints a and b, then reports the largest value as the absolute maximum and the smallest as the absolute minimum for that closed interval.