Engineering Science 1036A/B Study Guide - Final Guide: Cube Root, Negative Number, Coefficient
29 Jan 2020
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Department
Professor
Multiple Choice Tips
Question:
Given the following information, which of the following statements is false?
f(x) = 3x4 + 4x + 8
f’(x) has only one x-intercept
f’’(0) = 0
A. f(x) has a point of inflection at x = 0
B. f(x) has a critical point at x = -0.69
C. f’(x) has a degree of 3
D. f’’(x) > 0 x∈R
Possible Answer
Rationale
A. f(x) has a
point of
inflection
at x = 0
Correct Answer:
f(x) = 3x4 + 4x + 8
f’(x) = 12x3 + 4
f’’(x) = -36x2
Set f’’(x) = 0
0 = 36x2
0 = x
x = 0
x < 0
x = 0
x > 0
Test
Value
-1
0
1
f’’(x)
+
0
+
f(x)
Since
f’’(x) > 0
Therefore
it is
Concave
up
(0,8)
Since
f’’(x) > 0
Therefore
it is
Concave
up
Since there is no change in concavity on either side of f(x), therefore there is no
point of inflection at (0,8).
There are two requirements for a P.O.I, 1) the second derivative must equal
zero, 2) there must be a change in sign on either side of the point where it
equals zero on the second derivative. There must be a change in concavity for
f(x). The second derivative test is required. In this situation, f’’(0) = 0, however
there second derivative is always positive, there is no change in sign. A student
may forget this fact and automatically assume that there is a P.O.I at x = 0 solely
on the fact that f’’(x) equals to zero. This will cause them to assume that this
statement is true instead of false.
Document Summary
Possible answer: f(x) has a point of inflection at x = 0. Correct answer: f(x) = 3x4 + 4x + 8 f"(x) = 12x3 + 4 f""(x) = -36x2. Since there is no change in concavity on either side of f(x), therefore there is no point of inflection at (0,8). There are two requirements for a p. o. i, 1) the second derivative must equal zero, 2) there must be a change in sign on either side of the point where it equals zero on the second derivative. There must be a change in concavity for f(x). In this situation, f""(0) = 0, however there second derivative is always positive, there is no change in sign. A student may forget this fact and automatically assume that there is a p. o. i at x = 0 solely on the fact that f""(x) equals to zero.
