The number of distinct real roots of `x^4-4x^3+12 x 62+x-1=0i s________dot`

Options

A. 1
B. 0
C. 2
D. 4

Answer

C

Explanation

Let `f(x) = x^(4) - 4x^(3) + 12 x^(2)+ x - 1`.
Clearly, it is an even degree whose last term is negative and coefficient of highest degree term is positive. So, it has at least two real roots one positive and one negative.
We have, `f(0) = - 1, f(1) = 9 and f(-1) = 15`
Thus, one real root lies between 0 and 1, and other lies between -1 and 0.
Now,
`f'(x) = 4x^(3) - 12x^(2) + 24x + 1 and f''(x) = 12 (x^(2) - 2x + 2)`
Clearly, f''(x) has no real root and f'(x) being and odd degree polynomial, has at least one real root. So, by the algebraic interpretation of Rolle's theorem f'(x) has exactly one real root. Consequently, f(x) has exactly two real roots.