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Home Continuous Functions Derivatives and Curve Sketching Theorem 1: Graphical Meaning of the First Derivative
 
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Theorem 1: Graphical Meaning of the First Derivative

THEOREM 1

Suppose f is continuous on I and has a derivative at every interior point of I.

(i) If f'(x) = 0 for all interior points x of I, then f is constant on I.

(ii) If f'(x) > 0 for all interior points x of I, then f is increasing on I.

(iii) If f'(x) < 0 for all interior points x of I, then f is decreasing on I.

A proof will be given in the next section.
Home Continuous Functions Derivatives and Curve Sketching Theorem 1: Graphical Meaning of the First Derivative

Last Update: 2006-11-05