If $f\left( x \right) = A\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right) + B,\;\,\,f'\,\,\left( {\frac{1}{2}} \right) = \sqrt 2 $ and $\mathop \smallint \limits_0^1 \,f\left( x \right)dx = \frac{{2\,A}}{\pi }\,$, then the constants $A$ and $B$ are respectively

$\frac{4}{\pi }$ and ${\rm{0}}$

$\frac{\pi }{2}$ and $\frac{\pi }{2}$

$\frac{2}{\pi }$ and $\frac{3}{\pi }$

${\rm{0}}$ and $ - \frac{4}{\pi }$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Medium

1 of 1

Application +4 | -1

f\left( x \right) = A\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right) + B,\;\,\,f'\,\,\left( {\frac{1}{2}} \right) = \sqrt 2

and

\int \limits_0^1 \,f\left( x \right)dx = \frac{{2\,A}}{\pi }\,

, then the constants

A

and

B

are respectively

View answer and explanation

Correct option: D

\frac{4}{\pi }

and

\mathrm{0}

Option D (

\frac{4}{\pi }

and 0) is correct. By using the given derivative and integral conditions, we can solve for the constants A and B.