[tex]sinx+\sqrt3cosx \leq 2[/tex]
[tex]f(x) = sinx+\sqrt3cosx[/tex]
[tex]f'(x) = 0 => -\sqrt3sinx+cosx=0[/tex]
[tex]-\sqrt3sinx=-cosx | (-1) => \sqrt3sinx=cosx | :cosx[/tex]
[tex]\sqrt3tgx=1 => tgx = \frac{1}{\sqrt3}[/tex]
[tex]x=\frac{\pi}{6}; \frac{7\pi}{6}[/tex]
[tex]f''(x) = -\sqrt3cosx-sinx[/tex]
[tex]f''(\frac{\pi}{6}) = -\sqrt3\frac{\sqrt3}{2} -\frac{1}{2} < 0 =>[/tex] f are maxim la [tex]x = \frac{\pi}{6}[/tex]
Valoarea maxima pentru [tex]f(\frac{\pi}{6}) = \sqrt3* \frac{\sqrt3}{2} + \frac{1}{2} = 2[/tex]
