MAT 117 - Arizona State University



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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