Find the maximum and minimum value of f(x)= sinx+root 3 cosx

1.	Find the maximum and minimum value of f(x)= sinx+root 3 cosx
Answer» GIVEN, $\longrightarrow\sf{f(x)=\sin x+\sqrt3\,\cos x}$ DIVIDING and multiplying 2 on RHS, $\longrightarrow\sf{f(x)=2\left[\dfrac{\sin x+\sqrt3\,\cos x}{2}\right]}$ $\longrightarrow\sf{f(x)=2\left[\dfrac{1}{2}\,\sin x+\dfrac{\sqrt3}{2}\,\cos x\right]}$ $\longrightarrow\sf{f(x)=2\left[\cos60^o\sin x+\sin60^o\cos x\right]}$ $\longrightarrow\sf{f(x)=2\left[\sin x\cos60^o+\cos x\sin60^o\right]}$ SINCE $\sf{\sin A\cos B+\cos A\sin B=\sin(A+B),}$ $\longrightarrow\sf{f(x)=2\sin(x+60^o)}$ Well, since $\sf{\sin x\in[-1,\ 1],}$ $\longrightarrow\sf{\sin(x+60^o)\in[-1,\ 1]}$ $\longrightarrow\sf{2\sin(x+60^o)\in[-2,\ 2]}$ That is, $\longrightarrow\sf{f(x)\in[-2,\ 2]}$ Hence, $\longrightarrow\sf{\underline{\underline{\min(f(x))=-2}}}$ $\longrightarrow\sf{\underline{\underline{\max(f(x))=2}}}$

Answer»

$\longrightarrow\sf{f(x)=\sin x+\sqrt3\,\cos x}$