Derivatives 1

1) f'(x) = 5x4 - 6 < 0 for all x in [-1, 1]

Hence f(x) is decreasing in -1 <= x <= 1 and therefore the curve y = f(x) can only cross the x-axis at most once.

So at most 1 real root in [-1. 1].

2) Let the roots be x1 and x2 where x1 =/= x2, then f(x1) = f(x2) = 0.

By MVT, there exists c between x1 and x2 such that:

f'(c) = [f(x1) - f(x2)]/(x1 - x2) = 0

Hence f'(x) = 0 has at least one root.