Question

Sa se calculeze valoarea maxima a expresiei:
a) E(x)=sinx cosx, x∈R
b) F(x)=sinx+cosx , x∈R

Answer 1

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