Consider the function `f:(-oo,oo)vec(-oo,oo)` defined by `f(x)=(x^2+a)/(x^2+a),a >0,` which of the following is not true? maximum value of `f` is not attained even though `f` is bounded. `f(x)` is increasing on `(0,oo)` and has minimum at `,=0` `f(x)` is decreasing on `(-oo,0)` and has minimum at `x=0.` `f(x)` is increasing on `(-oo,oo)` and has neither a local maximum nor a local minimum at `x=0.`

Options

A. ` ( 2 + a )^(2) f '' (1) + ( 2- a ) ^(2) f'' ( - 1 ) = 0 `
B. ` ( 2 - a ) ^(2) f '' (1) - ( 2 + a ) ^(2) f '' ( - 1 ) = 0 `
C. ` f ' (1) f' (-1) = - ( 2 + a ) ^(2)`
D. `f ' (1) f ' (-1) = - ( 2 + a ) ^(2)`

Answer

a

Explanation

`f (x) = ((x^(2) + ax + 1) - 2 ax ) /(x^(2) + ax + 1) =1- (2ax)/(x ^(2) + a x + 1)`
`f' (x) = - [((x ^(2) + ax + 1) * 2a - 2 ax ( 2x + a))/( ( x^(2) + ax + a ) ^(2)) ] `
`" " = [ (- 2ax ^(2) + 2a )/((x ^(2) + ax + a ) ^(2)) ] = 2a [ ((x^(2) - 1))/( (x ^(2) + ax + 1)^(2)) ] ` ... (i)
` f'' (x) = 2 a [ ((x^(2) + a x + 1) ^(2) ( 2x) - 2 ( x^(2) - 1) (x ^(2) + a x + 1) ( 2x + a))/( (x ^(2) + ax + 1) ^(4)) ]`
` " " = 2a [ (2x ( x^(2) + a x + 1 ) - 2 (x^(2) - 1) ( 2x + a))/( (x^(2) + ax + 1) ^(3)) ] ` ... (ii)
Now, ` f'' (1) = ( 4 a (a + 2)) /( (a + 2) ^(3)) = ( 4 a ) /( ( a + 2) ^(3)) `
and ` f '' (-1) = ( 4a (a - 2 ) )/( ( 2- a ) ^(3)) = (- 4a )/( (a -2) ^(2))`
`therefore ( 2+ a ) ^(2) f '' (1) + ( 2 -a ) ^(2) f'' ( - 1) = 4a - 4a = 0 `