Let `(f(x+y)-f(x))/2=(f(y)-a)/2+x y` for all real `xa n dydot` If `f(x)` is differentiable and `f^(prime)(0)` exists for all real permissible value of `a` and is equal to `sqrt(5a-1-a^2)dot` Then `f(x)` is positive for all real `x` `f(x)` is negative for all real `x` `f(x)=0` has real roots Nothing can be said about the sign of `f(x)`

Options

A. f(x) is positive for all real x
B. f(x) is negative for all real x
C. f(x)=0 has real roots
D. nothing can be said about the sign of f(x)

Answer

A

Explanation

We have `(f(x+y)-f(x))/(2)=(f(y)-a)/(2)+xy`
`therefore" "f(x+y)=f(x)+f(y)-a+2xy`
Put x = y = 0
`rArr" "f(0)=a`
`f'(x)=underset(hrarr0)(lim)(f(x+h)-f(x))/(h)`
`=underset(hrarr0)(lim)(f(x)+f(h)-a+2xh-f(x))/(h)`
`=underset(hrarr0)(lim)(f(h)-a)/(h)+2x`
`=underset(hrarr0)(lim)(f(h)-f(0))/(h)+2x`
`=f'(0)+2x`
`rArr" "f'(x)=2x+sqrt(5a-1-a^(2))`
`rArr" "f(x)=x^(2)+(sqrt(5a-1-a^(2)))x+c`
`rArr" "f(x)=x^(2)+(sqrt(5a-1-a^(2)))x+c`
Putting x = 0, we get c = a
`rArr" "f(x)=x^(2)+(sqrt(5a-1-a^(2)))x+a`
`=(x+(sqrt(5a-1-a^(2)))/(2))^(2)+a-(5a-1-a^(2))/(4)`
`=(x+(sqrt(5a-1-a^(2)))/(2))^(2)+(4a-5a+1+a^(2))/(4)`
`=(x+(sqrt(5a-1-a^(2)))/(2))^(2)+(a^(2)-a+1)/(4)gt0`
`" "(as a^(2)-a+1gt0)`
`therefore" "f(x)gt0 AA x in R`