<s>[INST] To find maximum value If $A>0,B>0$ and $C>0$ and further it is known that $A+B+C=\frac{5\pi}{4}$,then find the maximum value of $\sin A+\sin B+\sin C$ 
 [/INST] From a purely algebraic point of view, elimating $C$ from the constraint leads to maximizing $$F=\sin \left(A+B-\frac{\pi }{4}\right)+\sin (A)+\sin (B)$$ Taking derivatives $$F'_A=\cos \left(A+B-\frac{\pi }{4}\right)+\cos (A)=0$$ $$F'_B=\cos \left(A+B-\frac{\pi }{4}\right)+\cos (B)=0$$ This implies $A=B$ and then $$\cos(2A-\frac{\pi }{4})+\cos(A)=2 \cos \left(\frac{A}{2} -\frac{\pi }{8}\right) \cos \left(\frac{3A}{2}
   -\frac{\pi }{8}\right)=0$$ and the only acceptable solution is $A=\frac{5 \pi }{12}$ with $B=C=A$ and then a maximum value equal to $$\frac{3 \left(1+\sqrt{3}\right)}{2 \sqrt{2}}$$
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