25: The derivative
f ′(x) = 4x3 + 4x − 1
may have one to three roots, it is not in any way special. Trying one more
derivative yields
f ′′(x) = 12x2 + 4.
As a positive function, this has no roots, consequently
f ′ can
have at most one root and f itself can
have at most two roots. The function goes to infinity at both infinity and
minus infinity, so it may have no, one or two roots. Since
the first derivative is not "nice", it is not easy to find critical points
and therefore investigation of monotonicity does not offer an easy way to
find out how many roots there actually is.
Thus it may pay off to just try to use the
Intermediate value
theorem to identify positions of potential roots, try substituting nice
integer values into f and wait for a sign change between two
successive ones; see how many times this happens.
Next hint
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