9853

A Generalization of the Mean Value Theorem

Theorem: Let be a function continuous on and differentiable on . Then there is a in such that .
Proof: the theorem follows by applying Rolle's theorem to the auxiliary function
Here is a geometric interpretation: The triangle formed by the axis, the tangent line through , and the secant line through and the point is an isosceles triangle (the green triangle). Therefore the slopes of the two sides not on the axis are and .
The example used is the function .

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Reference:
J. Tong, "The Mean-Value Theorem Generalised to Involve Two Parameters," Math. Gazette, 88(513), 2004 pp. 538–540.
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