(a) Let f(x) = x² and g(x) = −x³ + x² + 3x + 2. Then f(−1) = g(−1) and f(2) = g(2). Show that there is at least one value c in the interval (−1, 2) where the tangent line to f at (c, f(c)) is parallel to the tangent line to g at (c, g(c)). Identify c.

(b) Let f and g be differentiable functions on [a, b], where f(a) = g(a) and f(b) = g(b). Show that there is at least one value c in the interval (a, b) where the tangent line to f at (c, f(c)) is parallel to the tangent line to g at (c, g(c)).