Calculator Inputs
The page stays in a single stacked flow, while inputs use a responsive 3-column, 2-column, and 1-column grid.
Example Data Table
| Theorem | Sample Inputs | Left Side | Right Side | Relation | Interpretation |
|---|---|---|---|---|---|
| AM-GM | a = 4, b = 9, c = 16 | 9.6667 | 8.3203 | ≥ | Arithmetic mean exceeds geometric mean. |
| RMS-AM | a = 2, b = 5, c = 8 | 5.3852 | 5.0000 | ≥ | Root mean square is larger. |
| Cauchy-Schwarz | u = (1,2,3), v = (2,1,4) | 294 | 256 | ≥ | Product of squared lengths dominates the dot square. |
| Cauchy-Schwarz | u = (1,2,3), v = (2,1,1) | 84 | 49 | ≥ | Product of sums of squares dominates. |
| Jensen for x² | x = (1,3,5), w = (2,3,5) | 13.8 | 11.56 | ≥ | Weighted second moment exceeds squared weighted mean. |
| Young | x = 2, y = 3, p = 2 | 6 | 6.5 | ≤ | Conjugate exponent q becomes 2. |
| Bernoulli | x = 0.25, n = 4 | 2.4414 | 2.0000 | ≥ | Binomial growth stays above the tangent line. |
Formula Used
AM-GM: (a + b + c) / 3 ≥ (abc)1/3 for a, b, c ≥ 0.
RMS-AM: √[(a² + b² + c²) / 3] ≥ (a + b + c) / 3 for non-negative inputs.
Cauchy-Schwarz: (Σuᵢ²)(Σvᵢ²) ≥ (Σuᵢvᵢ)² for real vectors.
Jensen for x²: Σwᵢxᵢ² ≥ (Σwᵢxᵢ)² after normalizing non-negative weights to sum to 1.
Young: xy ≤ xp/p + yq/q where p > 1 and q = p/(p − 1).
Bernoulli: (1 + x)n ≥ 1 + nx for x ≥ −1 and integer n ≥ 0.
How to Use This Calculator
- Select the inequality theorem you want to check.
- Enter the numeric values required for that template.
- Choose the decimal precision for displayed results.
- Press Submit to place the proof summary above the form.
- Review assumptions, proof steps, and equality conditions carefully.
- Download the computed summary as CSV or PDF when needed.
FAQs
1. Can this calculator prove any arbitrary inequality?
No. It works with selected classical templates such as AM-GM, Cauchy-Schwarz, Jensen, Young, Bernoulli, and RMS-AM. It provides numeric verification and structured proof guidance for those families.
2. Why does the result mention assumptions?
Every inequality has conditions. Some require non-negative numbers, some need integer exponents, and others require normalized weights. The calculator checks those conditions before calling the proof valid.
3. What happens if assumptions are violated?
The page still computes the numeric expressions when possible, but it warns that the chosen theorem may not legally apply. That helps you spot invalid input cases quickly.
4. Does Jensen always need normalized weights?
Yes, the weighted form uses weights summing to one. This calculator accepts any non-negative weights with positive total, then normalizes them internally before comparison.
5. Why is there a gap or slack value?
The gap measures how far one side is from the bound. A zero gap usually indicates an equality case, while a larger gap means the inequality is comfortably satisfied.
6. Can negative numbers be used anywhere?
Yes, Cauchy-Schwarz accepts any real vector entries. Bernoulli allows x values at least −1. The other listed templates here are presented in their standard non-negative form.
7. What does the PDF export include?
The PDF export contains the theorem name, status, left side, right side, assumptions, proof steps, and equality note from the current computed result block.
8. How should students use this for study?
Try several input sets, compare equality cases, and match each result with the formula section. This turns the calculator into a quick revision tool for classical inequalities.