8.4.4 The Second Derivative Test

Question 1

The Second Derivative Test

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.
• 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.
• 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.

Question 2

note

Answer 2

• The second derivative test indicates whether a critical point is a maximum point or minimum point without making a sign chart for the first derivative.
• To use this test, you must set the first derivative equal to zero and find the x-values that make it equal to zero or undefined. These are the critical points.
• In order for a critical point to be a minimum, the graph must be concave up at that point, as shown on the far left. Therefore, if the second derivative is positive, the critical point is a minimum point.
• On the other hand, for a critical point to be a maximum, the graph must be concave down at that point. Therefore, if the second derivative is negative, the critical point is a maximum point.
• The third case occurs when the second derivative equals zero at the critical point. In this case the second derivative test fails. The critical point might correspond to a maximum point, a minimum point, or neither. To find out, you will need to make a sign chart for the first derivative.
• The second derivative test may not save you much work if you are trying to sketch the graph of a function. However, it may be useful if you are solving a problem involving maximization or minimization.

Question 3

Given that the function f(x) has a critical point at x= e^1.7 and the second derivative is f′′(x)=0, what can be said about f(x) at x=e^1.7?

Answer 3

You need more information to determine if the point is a maximum or minimum

Question 4

Given that the function f(x) has a critical point at x=π/2 and the second derivative is f′′(x)=−sin x, what can be said about f(x)at x=π/2?

Answer 4

The point is a maximum

Question 5

Given that the function f (x) has a critical point at x = 3 and the second derivative is f ″(x) = 2, what can be said about f (x) at x = 3?

Answer 5

There is a minimum point there.

Question 6

Given that the function f (x) has a critical point at x = −1 and the second derivative is f ″(−1) = −3, what can be said about f (x) at x = −1?

Answer 6

There is a maximum point there

Question 7

If f(x) has a critical point at x=5,f′(5)=0, and f′′(x)=3x−5, what can be said about the function at x=5?

Answer 7

f(x) has a minimum point at x=5

Question 8

Given that the function f(x)has a critical point at x=√2and that the second derivative is undefined at that point, what can be said about the function at x=√2?

Answer 8

No additional information can be determined.