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 the possible values of $m$ and $M$ are

Let f'(x) = 192x³/(2+sin⁴ π x) for all x ∈ R with f(1/2) = 0. If m ≤ ∫_(1/2)¹ f(x) dx ≤ M, then the possible values of m and M are $f'(x) = \dfrac{192x^3}{2+\sin^4 \pi x},\quad f\!\left(\dfrac12\right)=0$ $m \le \displaystyle\int_{1/2}^{1} f(x)\,dx \le M \Rightarrow ?$ $1.\ \text{bound } f' \text{ via } \sin^4 \in [0,1]$ $2.\ f(x) = \int_{1/2}^{x} f' \Rightarrow \text{bound } f$ $3.\ \int_{1/2}^{1} f \Rightarrow m,\ M$ $2 \le 2+\sin^4\pi x \le 3 \Rightarrow \dfrac13 \le \dfrac{1}{2+\sin^4\pi x} \le \dfrac12$ $64x^3 \le f'(x) = \dfrac{192x^3}{2+\sin^4\pi x} \le 96x^3$ $f(x) = \displaystyle\int_{1/2}^{x} f'(t)\,dt$ $16x^4 - 1 \le f(x) \le 24x^4 - \dfrac32$ $\int_{1/2}^{1}(16x^4-1)\,dx = 2.6,\quad \int_{1/2}^{1}(24x^4-\tfrac32)\,dx = 3.9$ $2.6 \le \displaystyle\int_{1/2}^{1} f(x)\,dx \le 3.9$ $I < 13,\quad I > \tfrac12,\quad I > 0$ $\Rightarrow (1),(2),(3) \text{ fail};\quad 1 \le I \le 12\ \checkmark$ $\boxed{m=1,\ M=12} \quad (4)$