Let f: R rightarrow R be a function defined b | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) First,observe that $f(x) + f(1-x) = \frac{4^x}{4^x+2} + \frac{4^{1-x}}{4^{1-x}+2} = \frac{4^x}{4^x+2} + \frac{4/4^x}{4/4^x+2} = \frac{4^x}{4^x+2}

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(B) First,observe that $f(x) + f(1-x) = \frac{4^x}{4^x+2} + \frac{4^{1-x}}{4^{1-x}+2} = \frac{4^x}{4^x+2} + \frac{4/4^x}{4/4^x+2} = \frac{4^x}{4^x+2} + \frac{4}{4+2 \cdot 4^x} = \frac{4^x}{4^x+2} + \frac{2}{2+4^x} = \frac{4^x+2}{4^x+2} = 1.$Let $I = \int_{f(a)}^{f(1-a)} x \sin^4(x(1-x)) dx.$ Using the property $\int_A^B g(x) dx = \int_A^B g(A+B-x) dx,$ and noting that $A+B = f(a) + f(1-a) = 1,$ we have:$M = \int_{f(a)}^{f(1-a)} (1-x) \sin^4((1-x)(1-(1-x))) dx = \int_{f(a)}^{f(1-a)} (1-x) \sin^4((1-x)x) dx.$Thus,$M = \int_{f(a)}^{f(1-a)} \sin^4(x(1-x)) dx - \int_{f(a)}^{f(1-a)} x \sin^4(x(1-x)) dx.$This implies $M = N - M,$ which simplifies to $2M = N.$Given $\alpha M = \beta N,$ we have $\alpha M = \beta (2M),$ so $\alpha = 2\beta.$Since $\alpha, \beta \in N,$ the smallest values are $\beta = 1$ and $\alpha = 2.$The value of $\alpha^2 + \beta^2 = 2^2 + 1^2 = 4 + 1 = 5.$

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

Similar Questions

The value of $\int_{4}^{7} \frac{(11-x)^{2}}{x^{2}+(11-x)^{2}} d x$ is

If the integral $\int_{0}^{10} \frac{[\sin 2 \pi x ]}{ e ^{ x -[ x ]}} dx =\alpha e ^{-1}+\beta e ^{-\frac{1}{2}}+\gamma$,where $\alpha, \beta, \gamma$ are integers and $[ x ]$ denotes the greatest integer less than or equal to $x$,then the value of $\alpha+\beta+\gamma$ is equal to ........ .

$\int_{-5}^{5} \log \left(\frac{7-x}{7+x}\right) dx =$

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cos^{\frac{3}{2}} x}{\cos^{\frac{3}{2}} x + \sin^{\frac{3}{2}} x} \, dx = $ . . . . . . .

If $f(x) = \int\limits_0^x {2(\cos^2 3t + 3\sin^2 3t)dt}$,then $f(x + \pi)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module