Let f: R → R be a function defined by f(x) = 4^x / 4^x+2 and

Question

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

Answer 1

Correct answer: 5

\(\because f(x)=\frac{4^x}{4^x+2}\)

\(f(1-x)=\frac{4^{1-x}}{4^{1-x}+2} \)

\(f(x)+f(1-x)=1\)

\(\therefore f(a)+f(1-a)=1\)

\(M=\int\limits_{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x\)

\(M=\frac{1}{2} \int\limits_{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x\) (Using elimination of x)

\(M=\frac{N}{2} \Rightarrow 2 M=N \)

\(\alpha M=\beta N\)

\(\alpha = 2\ \& \ \beta= 1\)

\(\alpha ^2+ \beta^2= 4 +1=5\)