Question

Aflati valoarea maxima a expresiei E=sin(x/2)*cos (x/2) , unde x∈R

Answer 1

inmultesti  egalitatea  cu  2
2E=2sinx/28cosx/2
Aplici  formula  sin2α=2sinα*cosα
2E=sinx
Valoarea  maxima  a   lui   sinx este  1.Deci
2E=1=>
E=1/2

Answer 2

$E = sin\Big(\dfrac{x}{2} \Big) \cdot cos\Big(\dfrac{x}{2} \Big) = \dfrac{2\cdot sin\Big(\dfrac{x}{2} \Big) \cdot cos\Big(\dfrac{x}{2} \Big) }{2} = \dfrac{sin\Big( 2 \cdot \dfrac{x}{2} \Big)}{2} = \dfrac{sinx}{2} \\ \\ \\ -1\leq sinx\leq 1 \Big|\cdot \Big(\dfrac{1}{2} \Big) \Rightarrow -\dfrac{1}{2}\leq \dfrac{sinx}{2}\leq \dfrac{1}{2} \\ \\ \Rightarrow Valoarea maxima a lui E este \dfrac{1}{2}$