LMVT visually: Draw the secant line from (a,f(a)) to (b,f(b)). There exists at least one point c where the tangent line is parallel to this secant.

Rolle's visually: When the secant is horizontal (f(a) = f(b)), the tangent at c is also horizontal.

Multiple values of c: For f(x) = sin(x) on [0, 2pi], f(0) = f(2pi) = 0. f'(c) = cos(c) = 0 gives c = pi/2 and c = 3*pi/2. Rolle's guarantees at least one — there can be many.

The MVT does NOT say c is unique. It only guarantees existence. For monotone f' (concave or convex f), c is unique.