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MAT 210 Practice – 2nd Derivative Test for Extrema (plus).
1) If the first derivative of a function equals zero when x = c, what do we know?
a) A local maximum exists when x = c.
b) A local minimum exists when x = c
c) Either a or b
d) A horizontal tangent line exists at x = c
e) Frogs are Purple.
For problems 2 and 3, Assume that we are given a function, f(x),
whose 1st and 2nd derivatives are defined around x = c.
2) When x = c, if the first derivative is zero and the 2nd derivative is positive, what do we know?
3) When x = c, if the first derivative is zero and the 2nd derivative is negative, what do we know?
4) If f(x) has a local maximum when x = c, then which of the following are guaranteed to be true?
a) [pic]
b) [pic]is either zero or undefined
c) If [pic]is defined, then it is positive.
d) If[pic]is defined, then it is zero.
5) Let f(x) = x3 + 6x2 – 36x + 1
a) List all critical points, c, from the first derivative such that [pic]
b) For each of the values in part a, calculate [pic]
c) According to the second derivative test for extrema, what do the results mean?
6) Let f(x) = x3 + 3x2 + 3x + 3
a) List all critical points, c, from the first derivative such that [pic]
b) For each of the values in part a, calculate [pic]
c) According to the second derivative test for extrema, what do the results mean?
7) Let f(x) = xex
a) List all critical points, c, from the first derivative such that [pic]
b) For each of the values in part a, calculate [pic]
c) According to the second derivative test for extrema, what do the results mean?
Answers (Revision 1)
1) d
2) A local minimum exists at x = c
3) A local maximum exists at x = c
4a) false, the derivative could be undefined at that point (such as a cusp)
b) true
c) false, it would be negative
d) true
5a) [pic] when x = -6 or 2
b) [pic]= -24 [pic]= 24
c) Local maximum at x = -6 and a local minimum at x = 2
6a) [pic] when x = -1
b) [pic]= 0
c) Second derivative test is inconclusive
7a) [pic] when x = -1
b) [pic]
c) Local minimum at x = -1
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