QJEEM24Mab977: JEE Main 2024 Mathematics - Class 12 JEEM24Mab977 Solution: Let f : R → R be a function defined by f ( x ) = 4 x 4 x + 2…

Let $$f:\mathbb{R}\to \mathbb{R}$$ be a function defined by $$f(x)=\frac{4^x}{4^x+2}$$ and $$M=\underset{f(a)}{\overset{f(1-a)}{\int }}x{\sin }^4(x(1-x))dx,N=\underset{f(a)}{\overset{f(1-a)}{\int }}{\sin }^4(x(1-x))dx;a\ne \frac{1}{2}$$. If $$\alpha M=\beta N,\alpha ,\beta \in \mathbb{N}$$, then the least value of $${\alpha }^2+{\beta }^2$$ is equal to __________.