2026-09-29 9 min

Unconstrained Optimization Calculator

Free unconstrained optimization calculator: find stationary points of ax³ + bx² + cx + d via the derivative, with second-derivative classification.

Calculator

Cubic Coefficients

Coefficient a (x³)

Coefficient b (x²)

Coefficient c (x)

Constant d

Results update live as you type

Status:f(x) = 1x³ + -3x² + 0x + 00 (local maximum) · 2 (local minimum)

Solution Summary

Derivative disc.D = 36

Points(0, 0); (2, -4)

Naturelocal maximum + local minimum

Results

Optimal

First stationary point

(0, 0) — local maximum

f″ sign at the point

Second stationary point

(2, -4) — local minimum

f″ sign at the point

Derivative discriminant

Two stationary points (peak and valley)

Global min / max?

None — cubics are unbounded both ways

Only local extremes exist

Results are recomputed live and verified against the step-by-step work below.

Visualizations

Stationary values

0P1

-4P2

Classification

Two stationary points (peak and valley)100%

Formula

Without bounds, extremes can only hide where the slope vanishes — the derivative roots are the whole suspect list.

f′(x) = 3ax² + 2bx + c = 0 → stationary points; f″(x) = 6ax + 2b classifies

Alternative Form

f″ > 0 local min; f″ < 0 local max; f″ = 0 inflection (flat)

Variables

Symbol	Name	Unit	Description
a..d	Cubic coeffs	numbers	f(x) = ax³ + bx² + cx + d
f′	First derivative	quadratic	3ax² + 2bx + c
f″	Second derivative	linear	6ax + 2b classifier
points	Stationary set	points	0, 1, or 2 real locations

Explanation

Solve the quadratic derivative for stationary points, then sign the second derivative at each.

Derivation

Fermat: interior local extremes force f′ = 0; the second-derivative sign reveals bowl (min) versus cap (max).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the cubic.

f(x)

f:1x³ + -3x² + 0x + 0

Result: Function recorded

Step 2

Differentiate term by term.

f′(x) = 3ax² + 2bx + c

fp:3x² + -6x + 0

Result: Derivative ready

Step 3

Solve f′(x) = 0 with the quadratic formula.

Derivative roots

D:36

Intermediate: D = 36

Result: 2 stationary point(s)

Step 4

Classify with f″(x) = 6ax + 2b at each root.

Second derivative

n:2

Result: 0 → local maximum; 2 → local minimum

Step 5

Evaluate f at each stationary point.

Heights

n:2

Result: (0, 0); (2, -4)

Final Answer

(0, 0) — local maximum

Worked Examples

See how the calculator works with different scenarios.

Scenario

Classic two-point cubic.

Inputs

a:1

b:0

c:-3

d:0

Calculation

f′ = 3x² - 3 → x = ±1; f″ = 6x

Result

max (-1, 2); min (1, -2)

Interpretation

Negative second derivative at -1 (cap), positive at 1 (bowl).

Understanding Your Result

What This Means

The headline result is the exact answer for the unconstrained optimization calculator inputs you entered, with supporting values shown beside it.

Why It Matters

Homework, exams and real-world models all reduce to this calculation; a verified numeric answer beats guessing.

Professional Interpretation

Read the headline result first, then scan the supporting rows and the step list to confirm the arithmetic before reusing the value.

Recommendations

•Verify the headline result against the steps
•Test one worked example by hand
•Check edge cases separately
•Keep full precision until the final display
•Restate the answer in the problem context

Educational Content

Introduction

This unconstrained optimization calculator evaluates your inputs with closed-form arithmetic and shows every intermediate value.

Concept Explanation

Solve the quadratic derivative for stationary points, then sign the second derivative at each.

Applications

▸Homework and exams

▸Engineering checks

▸Finance models

▸Science labs

▸Programming logic

▸Everyday estimation

Benefits

•Instant verified answer
•Step-by-step work shown
•Charts that expose mistakes
•Handles decimals directly
•Free with examples

Limitations

•Numeric inputs only
•Real-number results
•Display rounding to 6 decimals

Best Practices

•Enter numbers carefully
•Check the steps line by line
•Verify with one example
•Watch units in word problems
•State the final answer with context

Common Misconceptions

Myth: One input is enough.

Truth: Most problems need every coefficient; missing values change the answer.

Myth: Rounding early is fine.

Truth: Keep full precision until the final display to avoid drift.

Myth: The chart is decoration.

Truth: The chart is a second computation — mismatched bars mean a bad input.

Myth: Any rearrangement works.

Truth: Only equivalent operations preserve equality; follow the steps.

Practical Tips

•Double-check each input
•Read the steps in order
•Compare against one worked example
•Verify edge cases separately
•Keep units consistent
•Use Reset to restore defaults
•Copy the summary for notes

Common Mistakes

Typing a value in the wrong field

Match each label to the formula variables before calculating.

Dividing by zero

Check denominators and leading coefficients first.

Ignoring the steps

The step list shows exactly where a sign or arithmetic slip hides.

Rounding mid-calculation

Round only the displayed answer.

Forgetting to verify

Substitute the answer back into the original statement.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers: Inputs and outputs are real numbers.

Exact inputs: Coefficients are taken as given.

Standard notation: Conventional symbols and operator precedence.

Display rounding: Six decimals in outputs.

Limitations

Be aware of these limitations when using this calculator:

Numeric scope: Symbolic parameters need the advanced pages.

Single scenario: One input set per calculation.

No units conversion: Convert units before entering values.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a stationary point?

Where f′(x) = 0: the graph levels out — a peak, valley, or flat terrace.

How does the second derivative classify?

f″ > 0 bowls upward (min), f″ < 0 caps downward (max), f″ = 0 is flat and needs a sign check.

Can a cubic have a global max?

No — odd-degree tails run to ±inf, so only local extremes exist.

Why differentiate?

Fermat: any interior local extreme must zero the slope, so the derivative roots are the complete suspect list.

What if D < 0?

The derivative never touches zero: the cubic climbs (or falls) everywhere with no flat spots.

How do I verify?

Nudge x slightly each way: a min rises both sides, a max falls both sides.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Showing your steps

Write one operation per line.

Checking answers

Substitute back every time.

Reading interval notation

Brackets include, parentheses exclude.

Word-problem setup

Name variables before computing.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Solution set: All values satisfying the given statement.
Interval notation: Compact bracket notation for ranges of the number line.
Critical value: A point where the expression is zero or undefined, splitting sign regions.
Verification: Substituting the answer back to confirm the original statement.
Closed form: An exact formula evaluated directly, not an approximation loop.
Edge case: Inputs where the usual rule breaks, such as zero denominators.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Math foundations Math is Fun — Index Purplemath — Lessons Wikipedia — Mathematics Cuemath — Math resources

Applicable Standards

Common Core HSA — creating and reasoning with equationsNCTM Algebra — representations and equivalence

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.