a) [tex]E(x)=\sin{x}\cos{x}=\frac{\sin{2x}}{2}[/tex]
Dar functia sinus are valoarea maxima 1 [tex]max(\sin{x})=1[/tex] pentru orice argument x
Atunci
[tex]max(E(x))=\frac{max(\sin{2x})}{2}=\frac{1}{2}[/tex]
b) Ridicam la patrat expresia
[tex]F^{2}(x)=(\cos{x}+\sin{x})^{2}=\cos^{2}{x}+\sin^{2}{x}+2\sin{x}\cos{x}=[/tex]
Stim ca in general
[tex]\cos^{2}{x}+\sin^{2}{x}=1[/tex] si facem maximum acestei formule
[tex]max(F^{2}(x))=max(1+2\sin{x}\cos{x})=max(1+2E(x))=1+2max(E(x))=1+2\frac{1}{2}=1+1=2[/tex] extragem radicam
[tex]max(F(x))=\sqrt{2}[/tex]
