IIT JEE , AIEEE Forum - Mathematics , Trignometry

hsbhatt

Related only to the post above:

When a function is concave, it has the property that

$f(w_1 x_1+w_2 x_2+w_3 x_3+...+w_n x_n) geq w_1f(x_1)+w_2f(x_2)+w_3f(x_3)+...+w_n f(x_n) \ \ ext{where} sum_{i=1}^n w_i = 1$

and all w_i are all positive

This is called Jensen's inequality. The inequality reverses if the function is convex.

In general, a function is concave if

$f(w_1 x_1+w_2 x_2) geq w_1 f(x_1)+w_2 f(x_2) ext {where} w_1+w_2 = 1$

If the function is differentiable, this condition translated to 'f"(x)

This is satisfied by sinx in the interval [0, rac{pi}{2}]

$ext{Hence} sin( rac{x_1+x_2+x_3}{3}) geq rac {sin(x_1)+sin(x_2)+sin(x_3)} {3}$

$x_1+x_2+x_3 = pi \ \ ext{Hence} sin(x_1) + sin(x_2) + sin(x_3) leq 3 sin ( rac{pi}{3}) = 3 rac{sqrt3}{2}$