Inequalities path · I13

Before this lesson: Weighted means, Mathematical language and sets

Your goal: Establish convexity on the relevant interval before applying Jensen.

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: Jensen inequality; convexity inequality.

Jensen’s inequality: the key idea

A function f is convex on an interval if its graph lies below every chord: f(tx+(1−t)y)≤tf(x)+(1−t)f(y) for 0≤t≤1. Jensen extends this to finitely many nonnegative weights summing to 1. The entire interval containing the inputs must be a convexity interval. For twice differentiable functions, f″≥0 is a sufficient test; one can also prove convexity algebraically without calculus. Strict convexity and positive weights force all inputs equal in the equality case. For concave functions the inequality reverses.

A worked example

Prove f(x)=x² is convex without calculus.

The chord value minus the function value is tx²+(1−t)y²−(tx+(1−t)y)²=t(1−t)(x−y)²≥0. It is positive for distinct x,y and 0<t<1.

Your turn: change one thing

For real a,b,c show a²+b²+c²≥(a+b+c)²/3 using Jensen.

Try this on paper before opening the explanation.

Compare your reasoning

Apply convex f(x)=x² with three weights 1/3: ((a+b+c)/3)²≤(a²+b²+c²)/3. Multiply by 3. Equality requires a=b=c.

Pause and check

A trap to avoid: Assuming convexity globally from a convenient-looking formula.

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 three-point equal-weight Jensen inequality from two-point convexity.

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

First combine x,y with weights 1/2,1/2. Then combine (x+y)/2 and z with weights 2/3,1/3. Convexity gives f((x+y+z)/3)≤(2/3)f((x+y)/2)+(1/3)f(z)≤[f(x)+f(y)+f(z)]/3.

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. For a convex function, the graph lies where relative to a chord?

Foundation

  1. Outside its domain
  2. Below or on it
  3. Above it always
  4. Only at its midpoint
Hint

Use the definition of convexity.

Answer and reasoning

Below or on it. The value at a weighted input is at most the corresponding weighted output.

2. For a concave function, Jensen’s inequality does what?

Foundation

  1. Has no weights
  2. Reverses
  3. Stays the same always
  4. Requires integer inputs
Hint

Apply convexity to −f.

Answer and reasoning

Reverses. Negating a concave function makes it convex and reverses the comparison.

3. Which function is convex on all real numbers?

Core

  1. −x²
  2. 1/x on all reals
  3. x\sqrt{x} on all reals
  4. x²
Hint

Use a squared-difference chord calculation.

Answer and reasoning

x². For x² the chord difference is t(1−t)(x−y)²≥0.

4. Real a+b+c=6. Minimum a²+b²+c²?

Core

  1. 18
  2. 36
  3. 12
  4. 6
Hint

Use Jensen for the square function.

Answer and reasoning

12. The sum is at least 6²/3=12, attained at a=b=c=2.

5. Why is checking a convexity interval necessary?

Stretch

  1. Intervals affect only notation
  2. A function can change from convex to concave
  3. Jensen requires whole numbers
  4. Every function is globally convex
Hint

The theorem must hold between all inputs.

Answer and reasoning

A function can change from convex to concave. Convexity on an unrelated interval does not justify a chord comparison for the chosen inputs.

6. Strict convexity with positive weights gives equality when?

Stretch

  1. Their sum is zero
  2. They are distinct
  3. One input is largest
  4. All inputs agree
Hint

A strict chord inequality rules out distinct positive-weight inputs.

Answer and reasoning

All inputs agree. Equality requires all positively weighted inputs to be equal.

7. Jensen’s weights are required to be what?

Foundation

  1. Nonnegative and summing to 1
  2. All negative
  3. Arbitrary with any sum
  4. Integers only
Hint

They form a weighted average.

Answer and reasoning

Nonnegative and summing to 1. These hypotheses keep the weighted input inside the interval.

8. For real a+b=10, minimum a²+b²?

Core

  1. 50
  2. 25
  3. 100
  4. 10
Hint

Apply convexity of x² to the average 5.

Answer and reasoning

50. (a²+b²)/2≥5², so the minimum is 50 at a=b=5.

9. Can two-point convexity be used repeatedly to derive finite Jensen?

Stretch

  1. Only for two equal inputs
  2. Only if all weights vanish
  3. Yes
  4. No
Hint

Combine groups of weighted inputs.

Answer and reasoning

Yes. Repeated weighted averaging extends the chord inequality to finitely many points.

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