If A, B, C are the angles of a triangle,then | vedclass

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(D) Let $f(x) = \sin x$ for $x \in (0, \pi)$.Since $f''(x) = -\sin x < 0$ for all $x \in (0, \pi)$,the function $f(x)$ is concave down.By Jensen's Ine

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$\frac{3}{2}$

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(D) Let $f(x) = \sin x$ for $x \in (0, \pi)$.Since $f''(x) = -\sin x By Jensen's Inequality,for a concave function,$\frac{f(x) + f(y) + f(z)}{3} \leq f\left(\frac{x+y+z}{3}\right)$.Let $x = A$,$y = B$,and $z = C - \frac{\pi}{2}$.Then $\frac{\sin A + \sin B + \sin(C - \frac{\pi}{2})}{3} \leq \sin\left(\frac{A + B + C - \frac{\pi}{2}}{3}\right)$.Since $A + B + C = \pi$,we have $\frac{\sin A + \sin B - \cos C}{3} \leq \sin\left(\frac{\pi - \frac{\pi}{2}}{3}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.Therefore,$\sin A + \sin B - \cos C \leq \frac{3}{2}$.The maximum value is $\frac{3}{2}$.

If $A, B, C$ are the angles of a triangle,then the maximum value of $(\sin A + \sin B - \cos C)$ is-

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