1. how do I show that cos(x^2) is greater than or equal to cos(x) for 0 < x < 1?
2. deduce that integral of cos(x^2)dx from 0 to pi/6 is greater than or equal to 1/2.
1) The $\displaystyle \cos$ is a decreasing function on $\displaystyle (0,1)~.$
If $\displaystyle x\in (0,1)$ then $\displaystyle x^2<x$ so $\displaystyle \cos(x^2)>\cos(x)~.$
2) There is a theorem: If each of $\displaystyle f~\&~g $ is integrable and $\displaystyle \left( {\forall x \in [a,b]} \right)\left[ {f(x) \leqslant g(x)} \right]$ then $\displaystyle \int_a^b f \leqslant \int_a^b g $.