Let f:R R be a function defined by f(x)=4^x4^x+2 and M= _f(a)^f(1-a)x ^4(x(1-x))dx,N= _f(a…

Question

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

Answer: 5

Accepted Answer

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

$M=\int _{f(a)}^{f(1-a)}(1-x)\cdot {\sin }^{4}x(1-x)dx$

$M=N-M\, 2M=N$

$\alpha =2;\beta =1$

Ans. 5

Solutions · 1