Inequalities path · I10

Before this lesson: Sum of squares, Arithmetic, geometric and harmonic means

Your goal: Recognise useful squares or denominators and state equality correctly.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Names you may know: Cauchy inequality; Cauchy–Bunyakovsky–Schwarz inequality; Titu’s lemma (Engel form).

Cauchy–Schwarz inequality: the key idea

Cauchy–Schwarz states (∑aibi)2≤(∑ai2)(∑bi2)(\sum a_{i}b_{i})^{2}\le (\sum a_{i}^{2})(\sum b_{i}^{2}) for real lists of equal length. Equality means the two vectors are linearly dependent; if one is the zero vector equality is automatic. For positive denominators yᵢ, substitute ai=xiyia_i=\frac{x_i}{\sqrt{y_i}} and bi=yib_{i}=\sqrt{y_{i}} to obtain Engel form ∑xi2yi≥(∑xi)2∑yi\sum\frac{x_i^2}{y_i}\ge\frac{(\sum x_i)^2}{\sum y_i}. The main skill is choosing the two lists, rather than naming the theorem. For two terms the difference is exactly (ad−bc)².

A worked example

For positive a,b,c with a+b+c=6, prove 1/a+1/b+1/c≥3/2.

Apply Engel form to numerators 1,1,1 and denominators a,b,c: the sum is at least 9/6=3/2. Equality requires a=b=c=2.

Your turn: change one thing

Prove (x+y+z)²≤3(x²+y²+z²) for real inputs.

Try this on paper before opening the explanation.

Compare your reasoning

Apply Cauchy to (x,y,z) and (1,1,1). Equality means x=y=z.

Pause and check

A trap to avoid: Dividing by zero in a ratio-based equality condition; use proportional vectors.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove the two-term Cauchy inequality and identify equality.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Expand (a²+b²)(c²+d²)−(ac+bd)²=a²d²+b²c²−2abcd=(ad−bc)²≥0. Equality is equivalent to ad=bc, including cases where a vector is zero.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. Which inequality is Cauchy–Schwarz?

Foundation

  1. (ac+bd)²≥(a²+b²)(c²+d²) always
  2. a+b≤c+d
  3. ab≤cd
  4. (ac+bd)²≤(a²+b²)(c²+d²)
Hint

Pair two real vectors.

Answer and reasoning

(ac+bd)²≤(a²+b²)(c²+d²). The square of their dot product is at most the product of their squared lengths.

2. Engel form requires denominators to be what?

Foundation

  1. Integers only
  2. Positive
  3. Negative
  4. Arbitrary including zero
Hint

Square roots and division occur in the substitution.

Answer and reasoning

Positive. Positive denominators make the stated inequality valid.

3. Positive a+b=8. Minimum of 1/a+1/b is what?

Core

  1. 2
  2. 8
  3. 1/2
  4. 1/4
Hint

Use Engel form with two ones.

Answer and reasoning

1/2. The sum is at least 4/8=1/2, attained at a=b=4.

4. What is the best constant k in (x+y)²≤k(x²+y²) for all reals?

Core

  1. 2
  2. 1
  3. 3
  4. 4
Hint

Use Cauchy and test x=y≠0.

Answer and reasoning

2. Cauchy gives 2; equality at x=y=1 forces k≥2.

5. For nonzero vectors, Cauchy equality occurs when they are what?

Stretch

  1. Perpendicular
  2. Both increasing
  3. Both integer-valued
  4. Proportional
Hint

The dot product has maximal absolute value.

Answer and reasoning

Proportional. Linear dependence, equivalently proportional vectors, is the equality condition.

6. Which two lists give ∑xi2yi≥(∑xi)2∑yi\sum\frac{x_i^2}{y_i}\ge\frac{(\sum x_i)^2}{\sum y_i}?

Stretch

  1. xᵢ and yᵢ
  2. xᵢ² and yᵢ²
  3. 1/xᵢ and 1/yᵢ
  4. xiyi\frac{x_i}{\sqrt{y_i}} and yi\sqrt{y_{i}}
Hint

Make each paired product xᵢ.

Answer and reasoning

xiyi\frac{x_i}{\sqrt{y_i}} and yi\sqrt{y_{i}}. The products sum to ∑xi\sum x_{i}, while squared lengths are ∑xi2yi\sum\frac{x_i^2}{y_i} and ∑yi\sum y_{i}.

7. Can Cauchy–Schwarz use negative real entries?

Foundation

  1. No
  2. Only one negative entry
  3. Only if all entries are integers
  4. Yes
Hint

The squared norms remain nonnegative.

Answer and reasoning

Yes. The standard real-vector theorem has no positivity restriction on entries.

8. Positive a+b+c=9. Minimum 1/a+1/b+1/c?

Core

  1. 1/3
  2. 9
  3. 1
  4. 3
Hint

Use Engel with three numerators equal to 1.

Answer and reasoning

1. The lower bound is 9/9=1, attained at a=b=c=3.

9. Why is k=3 sharp in (x+y+z)²≤k(x²+y²+z²)?

Stretch

  1. All inputs must be zero
  2. Cauchy gives only k=9
  3. x=y=z≠0 forces k≥3
  4. Every k works
Hint

Test the equality case.

Answer and reasoning

x=y=z≠0 forces k≥3. At x=y=z=1, the inequality becomes 9≤3k.

Choose your next step

Continue to Hölder’s inequality. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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