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

Correct Answer: 5

Given:

The functional equation is given by: \[ f(a) + f(1 - a) = 1 \] 

Task: We are asked to calculate the values of \( M \) and \( N \).

Expression for \( M \): We are given the integral expression for \( M \): \[ M = \int_{f(0)}^{f(1)} (1 - x) \sin^4(x(1 - x)) \, dx. \]

Symmetry Consideration: By observing the symmetry of the problem, we deduce the relationship between \( M \) and \( N \). From the symmetry, we find that: \[ M = N - M, \] which simplifies to: \[ 2M = N. \]

Given Values: We are also given the values \( \alpha = 2 \) and \( \beta = 1 \). The least value of \( \alpha^2 + \beta^2 \) is computed as: \[ \alpha^2 + \beta^2 = 2^2 + 1^2 = 4 + 1 = 5. \]

Answer 2

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:

\[ \alpha^2 + \beta^2 = 2^2 + 1^2 = 5 \]