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A Generalization of the Mean Value Theorem

Theorem: Let

be a function continuous on

and differentiable on

. Then there is a

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

(31 lines omitted)

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

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