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Exercise 1
Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)
Chapter 1: Differentiation; Section 3: Derivatives; page 2
Properties, Exercise 1
Question
Study and do approximate graphic presentation of the following function. The study should contain:
- Roots and infinities
- Increasing and decreasing intervals
- Stationary points and their characteristics
- Points of infinite derivative and their characteristics
- Inflection points
Parts
- The obvious root is x=0 .
- The other roots can be obtained (if they exist) by division of the polynomial by x:
and one obtains:
- The derivative is:
- The stationary points are obtained from the derivative:
which gives:
- The second derivative is:
- From part 5 one obtains that in part 4 the upper x corresponds to a maximum and the lower x to a minimum.
- From part 5, the second derivative vanishes at
which is an inflection point since the third derivative does not vanish.
- The function is
increasing for x<−2.8685
decreasing for −2.8685<x<−0.4648 and
increasing for x>−0.4648 .
Score
Parts 2 and 4 are worth two points each.
The rest (parts 1,3,5,6,7,8) are worth 1 point each.