<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] We can use the method of Lagrange Multipliers. Note that the constraint function is $$ g(A,B,C) = A+B+C $$ and the maximization function is $$ f(A,B,C) = \sin A + \sin B + \sin C. $$We can find $ \nabla f $ and $ \nabla g $ and we get $$ \begin {eqnarray*} \nabla g &=& \left< 1, 1, 1 \right>, \\ \nabla f &=& \left< \cos A, \cos B, \cos C \right>. \end {eqnarray*} $$
Then, setting $ \nabla f \propto \nabla g $, we get $ \cos A = \cos B = \cos C $. We can go back to the restrictions that $A,B,C>0$ and $A+B+C=\frac{5\pi}{4}$ and we see that the only solution here is $$ (A,B,C) = \left( \frac{5\pi}{12}, \frac{5\pi}{12}, \frac{5\pi}{12} \right), $$ at which $ f(A,B,C) = \boxed{\frac {3 \sqrt{2+\sqrt{3}}}{2}} $. $\blacksquare$
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