Discussion
For \( f(x) = x^{2} - 4x + 3 \) on \( [1, 3] \), the point \( c \) guaranteed by Rolle's theorem is
Answer: D
The endpoint values agree, \( f(1) = f(3) = 0 \), so the theorem applies. Setting the derivative to zero,
\[ f'(x) \;=\; 2x - 4 \;=\; 0 \quad \Longrightarrow \quad c \;=\; 2, \]
which lies inside the interval as the theorem promises. It is the vertex of the parabola, the one place where the tangent is horizontal
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