Infinitesimal Calculus by James M. Henle

Author:James M. Henle
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1979-06-26T16:00:00+00:00

Hence, by theorem 5.2, y = x3 + 2x2 + x + 1 assumes a minimum at some x in Find the x in at which the minimum is assumed. To one who knows no calculus this problem is extraordinarily difficult, but to us it is simple, for a differentiable function, at the point where it attains a minimum, should have a tangent with slope 0.

The derivative of x3 + 2x2 + x + 1 is 3x2 + 4x + 1, and this is 0 at

Of these two points, only is in the interval and this turns out to be the point at which the function assumes its minimum in that interval. (Consider, in contrast, the interval At what x does the function assume its minimum here?)

This technique, pioneered by Pierre de Fermat early in the seventeenth century, is crystallized in the following theorem :

THEOREM 7.7. Assume that f(x) is differentiable on the open interval (a, b) and that at a point c in (a, b), f assumes a maximum (or a minimum). Then f’(c) = 0.

PROOF: Let us assume f(c) is a maximum (a similar proof applies if it’s a minimum). Since f(c) is a maximum,

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