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where £ lies between x° and x.

Theorem 11.21

If y = /(x) is a strictly concave function, then using the first-order total differential to estimate the change in the function value caused by moving away from x° to any other point x always leads to an overestimate. That is, for Ay s /(x) - /(x°) we have that

Ay = dy(x°) + ]-d2y($) Ay <dy{\°) since d2y < 0 for y = /(x) strictly concave.

Theorem 11.22

1 f v = / (x) is a strictly convex function, then using the first-order total differential

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