Question

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x) = \frac{4^x}{4^x + 2}$ and $M = \int_{f(a)}^{f(1 - a)} x \sin^4 \left( x (1 - x) \right) \, dx,$ $N = \int_{f(a)}^{f(1 - a)} \sin^4 \left( x (1 - x) \right) \, dx; \quad a \neq \frac{1}{2}.$ If $\alpha M = \beta N$, $\alpha, \beta \in \mathbb{N}$, then the least value of $\alpha^2 + \beta^2$ is equal to _____

Answer 1

Given:

$f(a) + f(1 - a) = 1$

Calculate $M$ and $N$:

$M = \int_{f(0)}^{f(1)} (1 - x) \sin^4(x(1 - x)) \, dx$

From symmetry, we have:

$M = N - M \quad \implies \quad 2M = N$

With $\alpha = 2$ and $\beta = 1$, the least value is:

$\alpha^2 + \beta^2 = 2^2 + 1^2 = 5$