What is a critical point?

A critical point of a function f is a value c in the domain where the derivative f prime of c is either zero or undefined. Every local maximum and local minimum of a smooth function is a critical point, which makes critical points the candidates to check when you are looking for extrema.

How does the second-derivative test work?

At a critical point c, evaluate the second derivative. If f double prime of c is positive, the curve is concave up at c, so the point is a local minimum. If f double prime of c is negative, the curve is concave down, so the point is a local maximum. If f double prime of c is zero, the test is inconclusive and you fall back to the first-derivative test or check higher derivatives.

What is the difference between a local and a global extremum?

A local maximum or minimum is the largest or smallest value of the function in some neighborhood around the point. A global maximum or minimum is the largest or smallest value over the entire domain or over a specified interval. On a closed bounded interval, the global extrema occur either at a critical point inside the interval or at one of the endpoints.

When does the second-derivative test fail?

The test fails when the second derivative at the critical point is zero, because the concavity of the curve at that point is not pinned down. The point could be a local max, a local min, or an inflection point with a horizontal tangent. In this case you fall back to the first-derivative test, which examines the sign of the first derivative on each side of the critical point.

Why does optimization rely on derivatives?

At a smooth maximum or minimum, the tangent line to the graph is horizontal, which means the derivative is zero at that point. Setting the derivative equal to zero gives a finite list of candidate points to check, turning the problem of finding the best value into an algebraic equation. The derivative is the bridge between the geometric idea of an extremum and a concrete computation.

Optimization Visualizer - Critical Points

x³ − 3xf(x) = x³ − 3x

880 × 460

f(x)f′(x)critical points (numbered)

Critical points on [a, b][a, b] = [-3, 3]

#

x

f(x)

f''(x)

Second-derivative test

Verdict

1

-1

2

-6

f′′(c) < 0 → concave down

local max

2

1

-2

6

f′′(c) > 0 → concave up

local min

f'(c) = 0 finds the candidates; sign of f''(c) classifies them. Positive: local min. Negative: local max. Zero or sign change: inflection.2nd-deriv test

Applieda=-3b=3f'=0 candidates · f'' classifies

Critical points of x³ − 3x

Critical points are the x values where

f'(x) = 0

(or undefined). They're the candidates for local max, local min, or inflection. The second derivative tells you which.

Derivatives ·

f(x) = x³ − 3x; f′(x) = 3x² − 3; f′′(x) = 6x

Right now

Found 2 critical points in

[-3, 3]

: 1) x = -1 (local max); 2) x = 1 (local min).

Reading the picture

The solid curve is

f

. The dashed curve is

f'

. At every critical point,

f'

crosses (or touches) zero. If

f'

goes from positive to negative there, you're at a local max of

f

. Touch-and-return (like

x^3

at 0) means the second-derivative test is inconclusive — use the first-derivative test.

Key Terms

Getting Started

The Function Families

The a and b Sliders

Reading the Critical-Point Table

Classification - Max, Min, Inflection

Display Toggles

What Is Optimization

The First-Derivative Test

The Second-Derivative Test

Related Concepts

Key Terms

Critical point — a value

c

in the domain of

f

where

f'(c) = 0

f'(c)

is undefined. Every local maximum or minimum of a smooth function is a critical point.

Local maximum — a point

c

where

f(c) \ge f(x)

for all

x

near

c

. The function value is at least as large as everything nearby.

Local minimum — a point

c

where

f(c) \le f(x)

for all

x

near

c

. The function value is at least as small as everything nearby.

Inflection point — a point where

f''

changes sign. The curve switches from concave up to concave down or vice versa. An inflection point may or may not be a critical point.

Concave up —

f''(x) > 0

. The graph cups upward like a bowl.

Concave down —

f''(x) < 0

. The graph cups downward like a dome.

Second-derivative test — at a critical point

c

f''(c) > 0

gives a local min,

f''(c) < 0

gives a local max,

f''(c) = 0

is inconclusive.

Getting Started

The page opens with the x³ − 3x family loaded on the interval

[-3, 3]

. You see:

• The solid blue curve of

f(x) = x^3 - 3x

.

• The dashed deep-blue curve of

f'(x) = 3x^2 - 3

.

• A faint blue band over

[a, b]

marking the search window.

• Two colored markers on

f

at the critical points: one deep-blue for the local max at

x = -1

and one bright-blue for the local min at

x = 1

.

Below the graph, the critical-point table lists each critical point with its coordinates, the value of

f''

at the point, the second-derivative test verdict, and a final classification tag.

To explore, drag the left endpoint a and right endpoint b sliders to change the search window, or switch families in the left panel to see different patterns of extrema and inflection points.

The Function Families

Six families are organized into two groups in the left panel, each tagged with its characteristic extrema count.

Polynomial:

• Quadratic

f(x) = x^2

— exactly one local minimum at the origin. The simplest non-trivial optimization.

• x³ − 3x — a cubic with a local max and a local min. The canonical *both kinds at once* example.

• x³ — a critical point at the origin where

f'

touches zero but doesn't change sign. The second-derivative test fails; first-derivative test reveals an inflection.

• x⁴ − 4x² — a *W shape* with two local minima and a local maximum. Multiple extrema of mixed kinds.

Transcendental:

• sin(x) — periodic, with infinitely many extrema. On

[-2\pi, 2\pi]

you get four.

• e^(-x²) — the Gaussian bell curve. A single global maximum at the origin, flanked by two inflection points.

The a and b Sliders

The left endpoint a and right endpoint b sliders set the search window. As you drag:

• The faint blue band over

[a, b]

resizes.

• The critical-point finder re-runs and rebuilds the table.

• Critical points that fall outside the new window disappear; ones that fall inside appear.

• Markers, drop lines, and table rows update together.

If you set

a > b

, the tool silently swaps them — only the interval matters. If the interval shrinks below a tiny threshold, the table reports *interval too narrow* until you give it a real interval.

This is useful for two reasons: you can zoom in on a single critical point to study it in isolation, or you can widen the window to see how many extrema a family produces over a long range.

The Reset button next to Parameters returns the interval to the family's default.

Reading the Critical-Point Table

The boxed table below the graph has one row per critical point. Each row reports:

• # — a numbered marker matching the colored dot on the graph.

• x — the location of the critical point.

• f(x) — the function value at that point.

• f''(x) — the value of the second derivative, the key input to the classification test.

• Second-derivative test — a short phrase explaining the verdict, including the rare cases where the test fails and the tool falls back to the first-derivative test.

• Verdict — a colored tag: local max (deep blue), local min (bright blue), or inflection (light blue).

When the table is empty, the cause appears in italics — either no critical points exist in the interval, or the interval is too narrow.

Classification - Max, Min, Inflection

Every smooth critical point has

f'(c) = 0

. The sign of

f''(c)

tells you which kind:

• f''(c) > 0 — concave up at

c

, so the curve cups upward and

c

is a local minimum.

• f''(c) < 0 — concave down at

c

, so the curve cups downward and

c

is a local maximum.

• f''(c) ≈ 0 — the test is inconclusive. The tool falls back to the first-derivative test: look at the sign of

f'

on each side of

c

:

– Negative-then-positive: local minimum.

– Positive-then-negative: local maximum.

– No sign change: inflection with horizontal tangent, as in

f(x) = x^3

x = 0

.

The fallback test handles the cases where the second derivative is too gentle to give a clean verdict, but the first derivative still reveals what's happening at the critical point.

Display Toggles

The Display section in the left panel hides individual layers when one is in the way:

• f(x) — toggles the function curve.

• f'(x) — toggles the dashed derivative curve. Useful for verifying that

f'

crosses zero at every marked critical point.

• f''(x) — toggles the dotted second-derivative curve. Off by default to avoid clutter; turn on when working through the second-derivative test in detail.

• critical points — toggles the markers and drop lines for every critical point in the interval.

Any combination is valid. The legend below the graph updates to show only the visible layers.

The Accent color picker at the bottom recolors the live highlight throughout the tool.

What Is Optimization

Optimization in single-variable calculus means finding the maximum or minimum value a function takes — either over its entire domain (a *global* extremum) or over a restricted interval (a *local* extremum).

The basic idea: at a smooth extremum, the tangent line is horizontal, so the derivative is zero. Solving

f'(c) = 0

finds the candidates; the second-derivative test (or the first-derivative test as backup) classifies each one.

For an interval

[a, b]

, the absolute maximum and minimum live either at a critical point inside the interval or at one of the endpoints. The standard recipe: find every critical point in

(a, b)

, evaluate

f

at each one and at the two endpoints, and pick the largest and smallest values.

For deeper coverage of optimization problems and applications, see the optimization page.

The First-Derivative Test

Once you have a critical point

c

, the first-derivative test classifies it by looking at the sign of

f'

on each side:

•

f'

changes from positive to negative at

c

→ local maximum. The function was increasing, then started decreasing.

•

f'

changes from negative to positive at

c

→ local minimum. The function was decreasing, then started increasing.

•

f'

does not change sign at

c

→ not an extremum. Typically an inflection point with a horizontal tangent, like

f(x) = x^3

x = 0

.

The first-derivative test always works (as long as you can determine the sign of

f'

in small neighborhoods on each side), but it requires more bookkeeping than the second-derivative test. The tool uses it as a fallback when

f''(c) = 0

or is too close to zero to give a reliable verdict.

The Second-Derivative Test

The second-derivative test is the fast path: at a critical point

c

, look at

f''(c)

:

•

f''(c) > 0

→ concave up → local minimum.

•

f''(c) < 0

→ concave down → local maximum.

•

f''(c) = 0

→ inconclusive.

The intuition: concave-up means the curve cups like a bowl, so a horizontal tangent at the bottom of the bowl is a local minimum. Concave-down means it cups like a dome, so a horizontal tangent at the top is a local max.

When

f''(c) = 0

the test fails because the curve's concavity isn't pinned down at

c

— it could be a max, a min, or an inflection. Higher derivatives can sometimes resolve it (the third or fourth derivative test), but the cleaner fallback is the first-derivative test described above.

Related Concepts

Derivatives — the foundation. The fact that

f'(c) = 0

at a smooth extremum is what makes the whole technique work. See the derivative visualizer.

Critical points — full theory of critical points, including the case where

f'

is undefined.

Concavity and inflection points — the geometric companion to the second-derivative test. Where

f''

changes sign you get an inflection point of

f

.

Mean Value Theorem — guarantees that if

f' \ne 0

on an interval,

f

is strictly monotonic. Justifies many of the increasing/decreasing arguments around critical points. See the MVT visualizer.

Extreme Value Theorem — guarantees that a continuous function on a closed bounded interval attains its max and min. The reason endpoint checks are part of the optimization recipe.

Applied optimization — real-world setup of optimization problems: largest box from a given sheet, shortest path, cheapest cost. The calculus is always the same; the modeling step is the hard part.

Visual tools for calculus — limits, continuity, derivatives, FTC, MVT, Riemann sums.