Cauchy with Parameters · Little Blue Book, Visualized

What this move does

For any positive numbers $λ_{1}, \dots, λ_{n}$ , apply the Cauchy inequality to $λ_{i} a_{i}$ and $\frac{b _{i}}{λ _{i}}$ :

(\sum a_{i} b_{i})^{2} \leq (\sum λ_{i} a_{i}^{2}) (\sum \frac{b _{i}^{2}}{λ _{i}}),

with equality if and only if $λ_{i} a_{i}$ and $b_{i}$ are proportional. The $λ_{i}$ are “weights” we put in ourselves. They do not change whether the inequality is true, but they change what the right side looks like and where equality holds.

In the expression to be estimated, split each term into two factors and insert an undetermined $λ$ .
Choose $λ$ so that one factor on the right does not depend on the variable (or is exactly the given condition).
Check that the equality condition can then be met inside the domain. If it can, the bound is the extreme value.

Predict first

$- 1 \leq x \leq \frac{5}{2}$ . What is the maximum of $f (x) = x + 1 + 5 - 2 x$ ?

14

\frac{42}{2}

3

7

Choose an answer first; then you can see the result.

Work through one

Example: find the maximum of f(x) = √(x+1) + √(5−2x)

Step 1 / 4

No parameter: λ = 1
Applying Cauchy directly gives $f (x)^{2} \leq 2 (6 - x)$ . The dashed curve is the square root of the right side. It lies above $f$ everywhere and touches $f$ at $x = \frac{4}{3}$ , but it slopes down to the right and reaches $14$ at the left endpoint.

$f (x)^{2} \leq (1 + 1) [(x + 1) + (5 - 2 x)] = 2 (6 - x)$
Insert λ
Make x cancel
The touching point is the highest point

In general, for $f (x) = A x + h + B c - k x$ take $λ = k$ : $f (x)^{2} \leq (\frac{A ^{2}}{k} + B^{2}) (k h + c)$ . The job of the parameter is to balance the two expressions under the square roots by their coefficients of $x$ .

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: the upper bound from Cauchy is the maximum

$λ = 2$ : the upper bound is the constant $10.5 \approx 3.240$ , touching $f$ at its highest point.

Practice

Exercise 1Basic

$0 \leq x \leq 1$ . Find the maximum of $f (x) = x + 3 - 3 x$ .

Exercise 2Basic

Real numbers $x, y$ satisfy $x + y = 1$ . Find the minimum of $x^{2} + 2 y^{2}$ .

Exercise 3Advanced

Positive numbers $a, b, c$ satisfy $a + 2 b + 3 c = 6$ . Find the maximum of $(a + b + c)^{2}$ .

Exercise 4Challenge

$- \frac{1}{2} \leq x \leq 4$ . For which $x$ does $f (x) = 2 x + 1 + 2 4 - x$ reach its maximum?