1.

Let f(x)=ax^(2)+x+3andf(x)ge0AAx inR,AAainA" where "AsubR. "Also "L=Lim_(xto oo) (x+1-sqrt(ax^(2)+x+3)). Which one of the following statement is incorrect ?

Answer»

If L exist then a=1.
If L does not exist then range of a is `[(1)/(12),1)uu(1,oo)`.
`|f(X)|` is continuous and differentiable `AAx in R,AAainA`
`f(|x|)` is non-derivable at EXACT,y two points.

Solution :(i) `"As "ax^(2)+x+3ge0AAx inR`
`"So, "agt0andDiscle0rArr1-12ale0rArr1le12arArrge(1)/(12)`
`"So,"ain[(1)/(12),oo)`.
(II) `L=underset(xtooo)Lim(x+1-sqrt(ax^(2)+x+3))=L=underset(xtooo)Lim((x+1)^(2)-(ax^(2)+x+3))/((x+1)+sqrt(ax^(2)+x+3))=L=underset(xtooo)Lim((1-a)x^(2)(2a-1)x-2)/((x+1)+sqrt(ax^(2)+x+3))`
`rArrL={{:(oo_(,),if,ain[(1)/(12)_(,))),((1)/(2)",",if,a=1),(-oo_(,),if,ain(1_(,)oo)):}`
Now, werify alternatives.


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