Question

Given $f^{2}(x)+g^{2}(x)+h^{2}(x) \leq 9$ and $U(x)=3 f(x)+4 g(x)$ $+10 h(x)$, where $f(x), g(x)$ and $h(x)$ are continuous $\forall x$ $\in R$. If maximum value of $U(x)$ is $\sqrt{N}$, then find $N$.

Answer 1

Let $\vec{V}_{1}=3 \hat{i}+4 \hat{j}+10 \hat{k}$
and $\vec{V}_{2}=f(x) \hat{i}+g(x) \hat{j}+h(x) \hat{k}$
$U(x)=\vec{V}_{1} \cdot \vec{V}_{2}=\left|\vec{V}_{1}\right|\left|\vec{V}_{2}\right| \cos \theta$ $\leq\left|\vec{V}_{1}\right|\left|\vec{V}_{2}\right|$
$=\sqrt{9+16+100} \sqrt{f^{2}+g^{2}+h^{2}} \leq 3 \sqrt{125}=\sqrt{1125}$
$\therefore N=1125$