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Theorem 2: Graphical Meaning of the Second Derivative

The next theorem gives the geometric meaning of the sign of the second derivative.

THEOREM 2

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

(i) If f"(x) > 0 for all interior points x of I, then f is concave upward on I.

(ii) If f"(x) < 0 for all interior points x of I, then f is concave downward on I.

We have already explained the intuitive reason for Theorem 2. The proof is omitted.


Last Update: 2006-11-05