The Second Derivative Test (8.4.4)

Question 1

• If the graph of a function has a tangent line with a slope of 0 and the graph is concave up at the same point, then the point is a minimum point of the function. If the graph of the function is concave down at that point, then the point is a maximum point of the function.

Answer 1

• If the graph of a function has a tangent line with a slope of 0 and the graph is concave up at the same point, then the point is a minimum point of the function. If the graph of the function is concave down at that point, then the point is a maximum point of the function.

Question 2

• If the graph of a function has a tangent line with a slope of 0 and the second derivative is at that point is also 0, then the second derivative test is inconclusive.

Answer 2

• If the graph of a function has a tangent line with a slope of 0 and the second derivative is at that point is also 0, then the second derivative test is inconclusive.

Question 3

• The second derivative test states that if f (c) = 0 and the second derivative of f exists on an open interval containing c,
then f(c) can be classified as follows:
1) If f (c) > 0, then f(c) is a relative minimum of f.
2) If f (c) < 0, then f(c) is a relative maximum of f.
3) If f (c) = 0, then the test is inconclusive.

Answer 3

• The second derivative test states that if f (c) = 0 and the second derivative of f exists on an open interval containing c,
then f(c) can be classified as follows:
1) If f (c) > 0, then f(c) is a relative minimum of f.
2) If f (c) < 0, then f(c) is a relative maximum of f.
3) If f (c) = 0, then the test is inconclusive.