Rolle counting roots

• A.WrongBoundary and endpoint

  False

  because 'all antiderivatives' means the whole family F + C, and the root count changes with C.

  Take $f(x) = x^{3} + x$, whose only real root is 0 and is simple (SymPy: roots 0, ±i).

  Then $F = x^{4}/4 + x^{2}/2 + C$, and SymPy gives two real roots at C = −1, exactly one at C = 0, and none at C = +1.

  So 'exactly one' holds for a single value of C, not for all.

• B.WrongBase field or ring

  False.

  F is a quartic

  so it has four roots counted with multiplicity over $\mathbb{C}$ — but the question asks about real roots.

  At C = +1 the roots are all complex (SymPy returns $\pm\sqrt{2}/2 \pm i\sqrt{6}/2$, none real).

  Degree bounds the number of real roots; it never guarantees them.

• C.Correct

  True.

  F′ = f has exactly one real zero and changes sign there

  so F is strictly monotone on each side of that single turning point.

  A function that decreases then increases crosses any horizontal level at most twice

  so F has at most two real roots — and, as the worked example shows, 0, 1 and 2 all actually occur.

• D.WrongProperty not inherited

  False.

  It reads the root count of f across to F

  but a property of a function need not survive antidifferentiation.

  With $f = x^{3} + x$ and C = −1, SymPy gives F two real roots, $\pm\sqrt{\sqrt{5} - 1}$.

3 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.