Minimum of a Quadratic Function on a Hyperplane

Calculus Level 5

Consider the minimum of

$F = \sum_{i=1}^{10} { a_i x_i^2 }$

subject to

$\sum_{i=1}^{10} { b_i x_i } = c$

with $a_i = i$ and $b_i = i$ and $c = 10$

If the minimum is $F^*$ , find $100 / F^*$

The answer is 55.

1 solution

Otto Bretscher
Jan 18, 2016

By Cauchy-Schwarz we have $10^2=(\sum\sqrt{i}\times\sqrt{i}x_i)^2\leq \sum i\times \sum ix_i^2=55F$ ; equality is attained when $x_i=\frac{2}{11}$ for all $i$ . Thus $F^*=\frac{100}{55}$ and the answer is $\frac{100}{F^*}=\boxed{55}$

Minimum of a Quadratic Function on a Hyperplane

The answer is 55.

1 solution

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