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The same logic applies to figure m10 2b in b to the

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The same logic applies to Figure M.10-2(b). In (b), to the left of x = 2 (in Zone III), the slope is positive, but to the right of x = 2 (in Zone IV), it is negative. That is, the slope is steadily decreasing in value as x increases: the “slope of the slope,” the second deriva- tive, is negative . From Table M.10-2, we have that the second derivative of the function in (b) is a constant –2 < 0. If the second derivative of a function is negative at a point where the first derivative is zero, then at this point, the function has a (local) maxi- mum . One trick that may prove helpful in remembering the sign of the second derivative is the following. The graph in Figure M.10-2(a) may remind you of a grin (and a grin is a “positive” expression). The graph in Figure M.10-2 (b) may remind you of a frown (and a frown is a “negative” expression). The trick may seem silly, but it has the advan- tage that it is so silly that it is difficult to forget. Two final points should be noted, with the help of Figure M.10-3. First, we have used the expression “ local ” maximum and minimum. Point A in Figure M.10-3(a) is a local maximum point, since the value of the function at A is greater than for any points in the M10-6 MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER (b) y = x 3 27 x O D C y x (a) O A (c) y = x 3 O y x B y x FIGURE M.10-3
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immediate neighbourhood of A. Yet the value of the function at B is greater than at A (and in fact at any other point). B is therefore referred to as a global maximum . (Of course, any global maximum is also a local maximum.) Figure M.10-3(b) illustrates a case where point C is a local maximum and D is a local minimum. In Figure M.10-3(b), how- ever, there is no finite global maximum or minimum, because the function goes to minus infinity on the left and to plus infinity on the right. The second point is illustrated by Figure M.10-3(c). It is possible for the first deriva- tive of a function to be equal to zero at a point that is neither a maximum nor a mini- mum. At the origin, the derivative of the function y = x 3 is dy/dx = 3 x 2 = 3(0) = 0, but the origin is clearly not a local maximum or minimum. Indeed, the function y = x 3 has nei- ther a local maximum nor a local minimum at any point. Yet the origin is a significant point. The second derivative of the function (the derivative of the derivative) = 6 x . Hence its value is negative for x < 0, zero for x = 0, and positive for x > 0. This means that the function is becoming increasingly less steep as it approaches the origin from the left, and increasingly steeper as it moves away from the origin to the right. A point where this change in curvature occurs, such as the origin in Figure M.10-3(c), is called a point of inflection . Such inflection points (as shown in Figures 9-4 and 10-2 in the text) play an important role in the theory of production and costs. 2. Exercises 1. For the following functions, which have the form y = f ( x ), give the expressions for the first and second derivatives, and determine whether (and if so where) they attain local maxima or minima or inflection points. Calculate the values for y , dy/dx , and d 2 y / dx 2 at these points.
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