1.

Let f(x)=alphax^(2)-a+(1)/(x)"where"alpha is real constant. The smallest alphafor f(x)ge0"for all"xgt0 is-

Answer»

`(2^(2))/(3^(3))`
`(2^(3))/(3^(3))`
`(2^(4))/(3^(3))`
`(2^(5))/(3^(3))`

SOLUTION :`F(x)=alphax^(2)-2+(1)/(x)`
`f(x)=(alphax^(3)-2x+1)/(x)AAx(0,oo)`
`sophi(x)=AX^(3)--ax+1` should be positive
`phi(x)=ax^(3)-2x+1`
`phi^(')(x)=3alphax^(2)-2=0`
`x=+-sqrt((2)/(3alpha))`
Clearly `x=sqrt((2)/(3alpha))` point of minima
`phi(sqrt((2)/(3alpha)))ge0`
`sqrt((2)/(3alpha)){ALPHA.(2)/(3alpha)-2}+1ge0`
`sqrt((2)/(3alpha))(-(4)/(3))+1ge0`
`sqrt((2)/(3alpha))((4)/(3))lel`
`sqrt((2)/(3alpha))le(3)/(4)`
`(2)/(alpha)le(3^(2))/(4^(2))`
`alphage(32)/(27)`


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