Suppose f '' is continuous on (−∞, ∞).
(a)
If f '(1) = 0 and f ''(1) = −1,
what can you say about f ?
1)At x = 1, f has a local maximum.
2)At x = 1, f has a local minimum.
3)At x = 1, f has neither a maximum nor a minimum.
4)More information is needed to determine if f has a maximum or minimum at x = 1.
(b)
If f '(2) = 0 and f ''(2) = 0,
what can you say about f ?
1)At x = 2, f has a local maximum.
2)At x = 2, f has a local minimum.
3)At x = 2, f has neither a maximum nor a minimum.
4)More information is needed to determine if f has a maximum or minimum at x = 2.
(a)
4) More information is needed to determine if f has a maximum or minimum at x = 1. Because f''(1) = -1, it tells us that the graph of f is concave down at x = 1. However, without further information about the behavior of f' on either side of x = 1, we cannot determine whether there is a local maximum or minimum at that point.
(b)
3) At x = 2, f has neither a maximum nor a minimum. Since f''(2) = 0, it tells us that the graph of f is neither concave up nor concave down at x = 2. This means there is no local maximum or minimum at that point.
To answer these questions, we can use the Second Derivative Test. The Second Derivative Test states that if the second derivative of a function is continuous at a critical point, then the behavior of the function at that point can be determined based on the sign of the second derivative.
(a) First, let's consider the point x = 1. Given that f'(1) = 0 and f''(1) = -1, we have a critical point at x = 1.
Now, according to the Second Derivative Test, if f''(1) < 0, then f has a local maximum at x = 1. Since f''(1) = -1, which is less than 0, we can conclude that at x = 1, f has a local maximum. Therefore, the correct answer is (1) At x = 1, f has a local maximum.
(b) Similarly, let's consider the point x = 2. Given that f'(2) = 0 and f''(2) = 0, we have another critical point at x = 2.
Now, applying the Second Derivative Test, if f''(2) = 0, then the test is inconclusive, and we cannot determine whether f has a maximum or minimum at x = 2. Therefore, the correct answer is (4) More information is needed to determine if f has a maximum or minimum at x = 2.
(a) well, the graph is concave down, so ...
(b) consider x^3 and x^4