The equation `x^(3) - 6x^(2) + 15 x + 3 = 0` has

Options

A. only one positive root
B. two positive and one negative roots
C. no positive root
D. none of these

Answer

C

Explanation

Let `f(x) = x^(3) - 6x^(2) + 15x + 3`. Then, `f'(x) = 3x^(2) - 12x + 15 = 3(x^(2)-4x + 5)`
`rArr" "f'(x) = 3 {(x-2)^(2)+1} gt 0 "for all" x in R`
`rArr" "f(x)` is strictly increasing on R.
Also, `f(0) = 3 gt 0`.
Thus, `f(x) gt 0 "for all" x gt 0`.
Hence, f(x) has no positive root.