How many negative roots does the function \(f(x) = x^3 - 3x^2 + 2\) have?

To determine the number of negative roots of the function (f(x) = x^3 - 3x^2 + 2), we can use the concept of finding the roots by substituting negative values for (x) and investigating changes in the sign of (f(x)).

First, we evaluate the function at several negative points to observe any sign changes, which indicate the presence of roots.

1. When (x = -1):

[

f(-1) = (-1)^3 - 3(-1)^2 + 2 = -1 - 3 + 2 = -2

]

So (f(-1) < 0).

2. When (x = -2):

[

f(-2) = (-2)^3 - 3(-2)^2 + 2 = -8 - 12 + 2 = -18

]

Thus, (f(-2) < 0).

3. When (x = -0.5):

[

f(-0.5) = (-0.5)^3 - 3(-0.5)^2 + 2 = -