Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 20

Answer

They are all negative.

Work Step by Step

I draw a representation of the curves in x and y direction on the image below as seen for the side, to facilitate visualization. From the location of point P, horizontal can represent both x or y direction. First derivatives are the slope of the tangent line in respective direction. Drawing the tangent lines to the curves, and measuring the angle of inclination with respect to the horizontal plane and positive angles above horizontal, we can see that the angle of inclination is negative. Second derivatives represents are the rate of change of the slope of tangente line, if you would move point P to the right, the tangent line would be getting more vertical, therefore the angle b would increase in module, but get more negative, this means that the slope is decreasing, therefore its rate of change is negative. Or if you want to simplify things, the function is concave down.
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