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

Info icon This preview shows pages 6–8. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 6

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 7
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern