Question

In an acute angled triangle $ABC,\cos A+\cos B+\cos C\le \lambda ,$ then $\lambda$ is equal to

• A. $\frac{5}{2}$
• B. $\frac{3}{2}$
• C. $3$
• D. $2$

Answer 1

The correct option is B $\frac{3}{2}$

$0<A,B,C<\frac{\pi }{2}$ and $\cos x$ is concave down in $(0,\frac{\pi }{2})$
Let us consider $3$ Points $A,B,C$ on the curve.
$\Rightarrow$ Centroid of $△ABC$ will be at $G$
$Y-$coordinate of $P\ge Y-$coordinate of $G$
$\frac{\cos A+\cos B+\cos C}{3}\le \cos \left(\frac{A+B+C}{3}\right)=\frac{1}{2}$
$\{\because A+B+C={180}^{\circ }\}$
$\therefore \cos A+\cos B+\cos C\le \frac{3}{2}$