Mathematics

Problem

Let

f'(x) = \frac{192x^3}{2 + \sin^4 \pi x}

for all

x \in \mathbb{R}

with

f\left(\frac{1}{2}\right) = 0

. If

m \le \int_{1/2}^1 f(x)\,dx \le M

, then find the possible values of

m

and

M

. a.

m = 13, M = 24

m = \frac{1}{4}, M = \frac{1}{2}

m = -11, M = 0

m = 1, M = 12