If `a,` `b `, `c in R` and `3b^(2)-8ac lt 0`, then the equation `ax^(4)+bx^(3)+cx^(2)+5x-7=0` has a) all real roots b) all imaginary roots c)exactly two real and two imaginary roots d) none

Options

A. all real roots
B. all imaginary roots
C. exactly two real and two imaginary roots
D. none

Answer

C

Explanation

`(c )` `f(x)=ax^(4)+bx^(3)+cx^(2)+5x-7`
`f'(x)=4ax^(3)+3bx^(2)+2cx+5`
`f''(x)=12ax^(2)+6bx+2c=0`
`implies "disc"=36b^(2)-4(12a)(2c)`
`=12(3b^(2)-8ac) lt 0`
`(:' 3b^(2) lt 8ac)`
`f'(x)=0` has only one real root
`implies f(x)=0` has exactly two real roots.