Show that $\int_{0}^{\pi/6} {\cos (x^2)}\mathrm{d}x\ge\frac12$. 
Prove that $\displaystyle\int_{0}^{\frac\pi 6} {\cos ({x^2)}\mathrm{d}x\ge\dfrac12}$.

I know this is a Fresnel integral but without going into advanced calculus is there a way to show that this is true? using calculus 1 knowledge, I tried Riemann's sum to prove this and got stuck. Thanks for any help.

For $0 < x \le \frac \pi 6 < 1$ we have $x^2 < x$ and therefore
$$
 \int_{0}^{\pi/6} \cos (x^2) \, dx >  \int_{0}^{\pi/6} \cos (x) \ dx
 = \sin( \frac \pi 6) = \frac 12 
$$
