When you root your (in)equality

Algebra Level 4

− k ≤ x − 10 k \large -k~ \le ~ x ~ - ~10\sqrt[]{k}

Find the least value of x x which satisfies the equation for all positive real values of k k . ie k ∈ R + k \in \mathbb R^+ .


The answer is 25.

Viki Zeta by Viki Zeta , 19, India

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1 solution

Viki Zeta
Dec 9, 2016

− k ≤ x − 10 k u = k − u 2 ≤ x − 10 u 0 ≤ u 2 − 10 u + x Solving for u, at equation = 0 u = 5 ± 25 − x k = 5 ± 25 − x k ≤ 5 k ≤ 25 k ≤ 25 − x k ≤ 25 − x k ≤ 25 ∴ min(k) = 25 -k \le x - 10\sqrt[]{k} \\ u = \sqrt[]{k} \\ -u^2 \le x - 10u \\ 0 \le u^2 - 10u + x \\ \text{Solving for u, at equation = 0} \\ u = 5 \pm \sqrt[]{25 - x} \\ \sqrt[]{k} = 5 \pm \sqrt[]{25 - x} \\ \sqrt[]{k} \le 5 \\ k \le 25 \\ \sqrt[]{k} \le \sqrt[]{25 - x} \\ k \le 25 - x \\ k \le 25 \\ \color{#3D99F6}{\boxed{\therefore \text{min(k) = 25 }}}

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