Consequences of MVT

Real Analysis · Axiom Academy

LESSON Consequences of the Mean Value Theorem Four corollaries that turn a fact about one tangent line into global control of a function: zero slope means constant, the sign of the slope sets direction, equal slopes force a constant gap, and a bounded slope caps how much the function can move. 1. Recall: The Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b) , then there is some where the tangent is parallel to the secant joining the endpoints — the instantaneous rate equals the average rate. 2. Corollary 1: Zero Derivative ⟹ Constant If a function never changes ( f'=0 everywhere), it cannot drift — it must hold the same height across the entire interval. The animation slides a flat tangent along the graph: the slope is 0 at every point, so the curve stays pinned to one level. 3. Corollary 2: The Sign of f' Sets the Direction The monotonicity test. Where the tangent tilts up ( ) the function strictly rises; where it tilts down ( ) it strictly falls. The animation runs one point across a curve and tints the moving tangent green while it climbs and red while it descends — the tilt is the verdict. 4. Corollary 3: Equal Derivatives Differ by a Constant If two functions have the same slope at every point, they can never converge or spread apart — they ride along at a fixed vertical gap. The animation slides a point across two curves that share f'=g' ; the dashed gap between them stays the same height the whole way. This is exactly the +C in antiderivatives.

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