A First Course in Higher Algebra

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DIFFERENTIATION OF ALGEBRAIC FUNCTIONS 87 function is neither increasing nor decreasing, and the derivative is equal to zero. If a tangent to the curve be drawn at such a point, it will be parallel to the a;-axis. It does not necessarily follow that when the derivative is zero, the function will have a maximum or a minimum. The derivative of a^ + 1 (see Ex. 2, page 74) is zero when a; = 0, but this is a so-called point of inflection, and not a maximum or minimum of the function. If, at a certai
...n point, the derivative is zero, it can be determined whether the point is a maximum or a minimum, by investigating the sign of the deriva- tive for values of a; a little less and a little greater than the value which makes the derivative vanish. If the derivative is negative for a value a little less, and posi- tive for a value a little greater, than the critical value, the point in question is a minimum, since, the derivative being first negative and then positive, the function must first decrease and then increase.

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