Problem of the Week

Problem 6

$\sum_{k=1}^n A_k \text{ means $A_1 + ... + A_n$.}$

(i) Given that $a \neq 0$ , find an inequality, in terms of $a$ , $b$ and $c$ , that holds if and only if the quadratic $ax^2 + bx + c$ has at most one root.

(ii) Let $a_1, ..., a_n$ and $b_1, ..., b_n$ be arbitrary numbers. By considering $\sum_{k=1}^n (a_k x + b_k)^2$ as a quadratic in $x$ , show that

$\left(\sum_{k=1}^n a_k b_k\right)^2 \le \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right) \text{.}$

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Problem of the Week

Problem 6

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