Q:

Proving inequality with the mean value theorem for integrals

I'm really struggling to even make a start on this question and even just a hint to get me going would be much appreciated.

The function $f(x)$ is defined on $[0, \pi/2]$ as
$$f(x) =  \int_0^x \sin^{1/2}{t} \, dt$$
Show that
$$\frac{\pi}{4} \left(\frac{1}{2}\right)^{1/4} \lt \int_{\pi/4}^{\pi/2} \sin^{1/2}{t} \, dt \lt \frac{\pi}{4} $$

My understanding is that for $g(x) = \frac{df(x)}{dx}$ there exists some $c$ for which $g(c)$ is equal to the average value of $g(x)$ however I'm not entirely sure how I would then apply the average value formula to the function, as I have no idea how to (and don't think I'm supposed to) integrate $\sqrt{\sin{x}}$. Thanks in advance for any help.

A:

You should be aware that on $[\pi/4,\pi/2]$, $\sin{x}$ is increasing and has minimum value $1/\sqrt{2}$, maximum value $1$ (by evaluating at the endpoints). Then
$$ \left(\frac{1}{2}\right)^{1/4} < \sqrt{\sin{t}} < 1 $$
for every point in the interior of the interval of integration. Integrating this inequality from $\pi/4$ to $\pi/2$ gives
$$ \frac{\pi}{4}\left(\frac{1}{2}\right)^{1/4} < \int_{\pi/4}^{\pi/2} \sqrt{\sin{t}} \, dt < \frac{\pi}{4};   $$
the inequalities do not become equalities because the inequality is true for the entire interval of integration with only two exceptional points.

