Question

solve this question.​

Answer 1

$f(x) = \frac{ {x} }{ ({x}^{2} + {9}^{2}) ^{ \frac{3}{2} } } \: \: \: \: \: f( \frac{a}{ \sqrt{2} } ) = ?$

$\bold{We \: have \: to \: substitute \: x = \frac{a}{ \sqrt{2} } }$

$f( \frac{a}{ \sqrt{2} } ) = \frac{ \frac{a}{ \sqrt{2} } }{( \frac{a}{ \sqrt{2 + {a}^{2}} }) ^{ \frac{3}{2} } }$

$\frac{a}{( \frac{ {a}^{2} }{2} + {a}^{2} ) {}^{ \frac{3}{2} } }$

$= > \frac{\frac{a}{ \sqrt{2} } }{ \frac{( {a}^{2} + 2 {a}^{2} ) {}^{ \frac{3}{2} } }{2} } \: \: \: \: \: = > \frac{ \frac{a}{2 \frac{1}{2} } }{ \frac{(3 {a)}^{ \frac{3}{2} } }{2 \frac{3}{2} } }$

$f( \frac{a}{ \sqrt{2} } ) = \frac{a \times 2 \frac{3}{2} }{2 \frac{1}{2} (3 {a ^{2} )}^{ \frac{3}{2} } }$

$f( \frac{a}{ \sqrt{2} } ) = \frac{ a \times {2}^{1 + \frac{1}{2} } }{2 \frac{1}{2} \: 3 \frac{3}{2} \times {a}^{2} \times \frac{3}{2} }$

$f( \frac{a}{ \sqrt{2} } ) = \frac{a \times 2 \times 2\frac{1}{2} }{2 \frac{1}{2} \times 3 \times 3 \frac{1}{2} \times {a}^{3} }$

$f( \frac{ {a}^{2} }{ \sqrt{2} } ) = \frac{2a}{3 \sqrt{2 } {a}^{3} }$

$f( \frac{a}{ \sqrt{2} } ) = \frac{2a}{3 \sqrt{3} {a}^{3} }$

$f( \frac{a}{ \sqrt{2} } ) = \frac{2}{3 \sqrt{3} {a}^{3} }$