Question

How many negative roots will this function have?
f(x) = x7 – 2x4 + 7x2 + 2x – 2

Answer 1

x = -1 is a solution,
as -1 -2 + 7 -2 -2 = 0.

Answer 2

It's also evident that f'(x) is positive at x = -1, as
f'(x) = 7x^6 - 8x^3 + 14x + 2 and
f'(-1) = 7 + 8 - 14 + 2 = 3...note that this is POSITIVE.
Yet f(0) = -2, so there must be a root somewhere in BETWEEN
x = -1 and x = 0.
The function has to get back down across the x axis.

Answer 3

As to negative roots of magnitude greater than -1,
it's not possible, because the derivative will always be
positive for x < -1. The 7x^6 and -8x^3 terms (both positive)
in the derivative will always overwhelm the
14x term (only negative term) when x < -1.

Answer 4

So the answer is, two roots. One is -1, the other is between 0 and -1.