Consider the cubic equation `x^3 + ax^2 + bx + c = 0`, where `a, b, c` are real numbers, which of the following statements is correct?

Options

A. If `b^(2)-4acgt0` then `f^(-1)(0)` does not contain 0
B. If `b^(2)-4aclt0,` then `f^(-1)(0)` must contain 0
C. If `b^(2)-4acgt0`, then `f^(-1)(0)` may contain 0
D. If `b^(2)-4aclt0`, then `f^(-1)(0)` may contain 0

Answer

A

Explanation

As given `f(x)=ax^(2)+bx+c,"Let " y=f(x)` . To get `f^(-1)(x)` we express x is terms of y in equation `y=ax^(2)+bx+c`.
`(y)/(a)=x^(2)+(b)/(a)x+(c)/(a)=(x+(b)/(2a))-((b)/(2a))^(2)+(c)/(a)`
`x+(b)/(2a)=sqrt((y)/(a)+(b^(2))/(4a^(2))-(c)/(a))=(pmsqrt(4ay+b^(2)-4ac))/(2a)`
`x=(-bpmsqrt(4ay+b^(2)-4ac))/(2a)`
or, `f^(-1)(x)=(-bpmsqrt(4ax+b^(2)-4ac))/(2a)`
Putting x=0
`f^(-1)(0)ne0`. i.e. if `b^(2)-4acgt0`
`f^(-1)(0)` dose not contain 0, if `b^(2)-4acgt0`.