Final answer:
The statements about derivatives given to the student involve understanding the Mean Value Theorem and the conditions for a function to have a local extremum. Only statements II (related to the Mean Value Theorem) and III (concerning potential local extrema) are true. Statement I is false because a constant nonzero derivative can still yield equal function values at different points.
Step-by-step explanation:
The student's question involves calculus, specifically about the properties of differentiable functions and their derivatives. To answer each of the given statements:
Statement I: If f '(x) exists and is nonzero for all x, this does not automatically imply that f(1) ≠ f(0). This statement is false because f(x) could still be a linear function with a constant non-zero slope. In such a case, f '(x) would indeed be nonzero for all x, but f(1) could be equal to f(0) plus the slope.
Statement II: This is a restatement of the Mean Value Theorem (MVT) and is true. If f is differentiable between -1 and 1 and continuous on the closed interval [-1, 1], and f(-1) = f(1), then by MVT there must be some c in (-1, 1) such that f '(c) = 0.
Statement III: The fact that f '(c) = 0 implies only that c could be a point where f has a local maximum, a local minimum, or neither (it could be a point of inflection). So, this statement is true, as it says 'or' not 'and'.
The correct answer to the question is II and III only.