Discussion
Match the theorem with its conclusion. 1. Rolle 2. Lagrange 3. Cauchy P. A tangent parallel to the chord Q. A vanishing derivative R. A ratio of derivatives equal to a ratio of increments
Answer: C
Rolle's equal endpoints force a horizontal tangent, so the derivative vanishes. Lagrange drops that restriction and concludes only that some tangent matches the chord's slope. Cauchy compares two functions at once, equating the ratio of their derivatives to the ratio of their increments, and reduces to Lagrange when the second function is \( x \)
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