Let f: R R be a function defined by f(x)=4^x 4^x+2 and M= _ f(a) … - Vidyadip

MCQ

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

A
$4$
✓
$5$
C
$6$
D
$7$

✓

Answer

Correct option: B.

$5$

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

$M=\int_{f(a)}^{f(1-a)}(1-x) \cdot \sin ^4 x(1-x) d x$

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

$\alpha=2 ; \beta=1 ;$

Ans. $5$

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