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.

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