What is Fermat's Theorem?

Dear Student,

Let y= f(x) be a continuous function in a
closed interval [x1, x2] assuming its greatest (or least) value at an interior point e of that interval: x1< e < x2.

Then, if the derivative f’(x) of the function f(x) at the point e exists it is equal to zero: f’(e) = 0.

The geometrical meaning of Fermat’s theorem is: the tangent to the graph of a function at its highest (or lowest) point is parallel to the axis of the abscissas.

At the end point of the interval of definitions, the tangents are not parallel to the he axis of the abscissas.


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