Question

Determinați maximul expresiei $E(x)=(sinx+cosx)^2$ când x ∈ R

Answer 1

$\displaystyle E(x)=sin^2x+2sinxcosx+cos^2x=1+2sinxcosx=1+sin2x. \\ \\ Deoarece~sinA \in [-1;1]~\forall~A \in \mathbb{R},~rezulta~ca~E(x) \le 1+1=2. \\ \\ Maximul~lui~E~este~2~si~se~realizeaza~cand~sin2x=1,~adica \\ \\ pentru~2x=\frac{\pi}{2}+2k \pi \Leftrightarrow x= \frac{\pi}{4}+k \pi,~unde~k \in \mathbb{Z}.$