If-0-x-pi-2-Prove-that-2-pi-x-sin-x-x-without-graphical-method-

Question Number 110779 by ZiYangLee last updated on 30/Aug/20

If 0 ⩽ x ⩽ \frac{π}{2},

Prove that \frac{2}{π} x ⩽ \sin x ⩽ x

without graphical method .

Answered by Her_Majesty last updated on 30/Aug/20

f_{1} (x) = \frac{2 x}{π}

f_{1}^{'} (x) = \frac{2}{π}

f_{2} (x) = s i n x

f_{2}^{'} (x) = c o s x

f_{3} (x) = x

f_{3}^{'} (x) = 1

0 ⩽ x ⩽ \frac{π}{2} :

f_{1} (0) = f_{2} (0) = f_{3} (0) = 0

\land

f_{1}^{'} (x) ⩽ f_{2}^{'} (x) ⩽ f_{3}^{'} (x)

\Rightarrow

f_{1} (x) s t a y s ⩽ f_{2} (x)

f_{2} (x) s t a y s ⩽ f_{3} (x)

w i t h i n t h e g i v e n i n t e r v a l

Commented by ZiYangLee last updated on 31/Aug/20

Thanks!