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Let f(x)=alphax^(2)-a+(1)/(x)"where"alpha is real constant. The smallest alphafor f(x)ge0"for all"xgt0 is- |
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Answer» `(2^(2))/(3^(3))` `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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