Question

E(x)= sin x+cos x
Sa se arate ca E(x) nu poate fi = 2

Answer 1

$E(x) = \sin x+\cos x \\\\= \sin x + \sin\left(\dfrac{\pi}{2}-x\right)\\\\\\=2\sin \left(\dfrac{x+\dfrac{\pi}{2}-x}{2}\right)\cos \left(\dfrac{x-\left(\dfrac{\pi}{2}-x\right)}{2}\right) \\ \\\\=2\sin \dfrac{\pi}{4}\cos \left(x-\dfrac{\pi}{4}\right) \\ \\ =\sqrt{2}\cos \left(x-\dfrac{\pi}{4}\right)\\ \\\\\\-1\leq \cos \left(x-\dfrac{\pi}{4}\right)\leq 1\Rightarrow -\sqrt{2}\leq \sqrt 2 \cos \left(x-\dfrac{\pi}{4}\right)\leq \sqrt 2 \\ \\ \Rightarrow E(x) \in [-\sqrt{2},\sqrt{2}] \Rightarrow \boxed{E(x)\neq 2}$