If `f(x)=x^(3)+3x^(2)+4x+asinx+bcosx, forall x in R` is a one-one fuction, then the greatest value of `(a^(2)+b^(2))` is

Answer

1

Explanation

`f(x)` is one-one
`f^(')ge0AAxepsilonR`
`implies3x^(2)+6x+4+acosx-bsinxge0AAxepsilonR`
`implies3x^(2)+6x+4gebsinx-acosx AAxepsilonR`
`implies3x^(2)+6x+4gesqrt(a^(2)+b^(2))AAxepsilonR`
`implies3(x+1)^(2)+1gesqrt(a^(2)+b^(2))AAxepsilon`
`impliessqrt(a^(2)+b^(2))le1:.(a^(2)+b^(2))_(max)=1`