Question

The greatest value of $2\sqrt{\mathrm{x}}+3\sqrt{\mathrm{y}}+4\sqrt{\mathrm{z}}+5\sqrt{\mathrm{u}}$ for possible values of $\mathrm{x},\mathrm{y},\mathrm{z}$and $\mathrm{u}$ such that $4\mathrm{x}+9\mathrm{y}+16\mathrm{z}+25\mathrm{u}=720$ is

• A. $10\sqrt{5}$
• B. $12\sqrt{5}$
• C. $24\sqrt{5}$
• D. $36\sqrt{5}$

Answer 1

The correct option is C

$24\sqrt{5}$

The explanation for the correct answer.

Find the greatest value:

Given: $4\mathrm{x}+9\mathrm{y}+16\mathrm{z}+25\mathrm{u}=720=\mathrm{k}$ (constant)

Here $\mathrm{m}=\frac{1}{2},\mathrm{n}=\frac{1}{4}(0<\mathrm{m}<1)$

The greatest value of $2\sqrt{\mathrm{x}}+3\sqrt{\mathrm{y}}+4\sqrt{\mathrm{z}}+5\sqrt{\mathrm{u}}=4^{1-(½)}{\left(720\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

[Because the greatest value of ${{\mathrm{a}}_1}^{\mathrm{m}}+{\mathrm{a}}_2+\dots ..{{\mathrm{a}}_{\mathrm{n}}}^{\mathrm{n}}={\mathrm{n}}^{1-\mathrm{m}}{\mathrm{k}}^{\mathrm{m}}$ where ${\mathrm{a}}_1={\mathrm{a}}_2=\dots ..{\mathrm{a}}_{\mathrm{n}}$ and ${\mathrm{a}}_1+{\mathrm{a}}_2\dots ..+{\mathrm{a}}_{\mathrm{n}}=\mathrm{k}=$ constant, $0<\mathrm{m}<1$]

$$\begin{gathered} =\sqrt{4}\times \sqrt{720} \\ =2\times 12\sqrt{5} \\ =24\sqrt{5} \end{gathered}$$

Hence, option (c) is the correct answer.