1.

Consider f as a twice differentiable function such that f(x)+f^('')(x)=-xg(x)f^(')(x)AAxge0 where,g(x)ge0AA x ge0, then (AAxge0)

Answer»

`(f(x)^(2))+(f^(')(x)^(2))` is a non increasing FUNCTION
`(f(x))(2)LT3(f(0))^(2)+(2f^(')(0))^(2)`
`|f(x))ge alpha, alpha` is a fixed real constant
`lim_(xto oo) f(x).sin(1/x)` exist

Solution :`d/(DX)((f(x))^(2))=2f^(')(x).f(x)+2f^(')(x).f^(")(x)=-2xg(x)(f^(')(x))^(2)le0`


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