Upper bound on Sum of square roots

Question

Let $k_1,k_2\ldots k_t$ be integers and $\sum_{i=1}^{t}{k_i}=k$ where $k$ is fixed. What is the maximum value of $\sum_{i=1}^{t}\sqrt{k_i}$.

Answer 1

As mentioned, this is a direct application of Cauchy-Schwarz inequality.

Hint: consider $v=(1,\dots,1)$ and $w=(\sqrt{k_1},\dots,\sqrt{k_t})$ and use $$ |\langle v,w \rangle|^2\le\|v\|_2\|w\|_2 $$