First slide

Extreme values and Periodicity of Trigonometric functions

Question

$If A, B, C \in (- \frac{π}{2} \frac{π}{2}), then max value of \cos A + \cos B + \cos C$

Moderate

A

$\frac{1}{2}$
B

$\frac{3}{2}$
C

$\frac{1}{3}$
D

$\frac{1}{8}$

Solution

For the function which is concave downwards.
$\begin{array}{l} f (\frac{x_{1} + x_{2} + x_{3}}{3}) > \frac{f (x_{1}) + f (x_{2}) + f (x_{3})}{3} \\ \Rightarrow \frac{\cos A + \cos B + \cos C}{3} \leq \cos (\frac{A + B + C}{3}) \\ \Rightarrow \frac{\cos A + \cos B + \cos C}{3} \leq \frac{1}{2} \\ \Rightarrow \cos A + \cos B + \cos C \leq \frac{3}{2} \end{array}$

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