Question

The graph of f"(x), the second derivative of the continuous function f(x), is shown above on the interval a,b. On this interval, f(x) has only one critical point, which occurs at c. Which of the following statements is true about the functionf(x) on the interval a,b?

a) f(x) has a local maxiμm at
b) f(x) has a local miniμm at
c)f(x) has an inflection point at
d) f(x) is constant on the interval a,b.

Answer 1

The second derivative test can be used to determine the behavior of the function on a given interval based on the graph of its second derivative. The options provided are analyzed using the second derivative test, and option c) appears to be the most likely, indicating the presence of an inflection point.

The behavior of the function f(x) on the interval (a, b) based on the graph of f''(x), we can use the second derivative test. The second derivative test states that:

If f'(c) = 0 and f''(c) > 0, then f(x) has a local minimum at x = c.

If f'(c) = 0 and f''(c) < 0, then f(x) has a local maximum at x = c.

If f''(c) = 0, the test is inconclusive.

Now, let's analyze the given options:

a) f(x) has a local maximum at c: This would be true if f'(c) = 0 and f''(c) < 0. However, the second derivative test doesn't specify the sign of f'(c), so this option is not necessarily true.

b) f(x) has a local minimum at c: This would be true if f'(c) = 0 and f''(c) > 0. Again, the second derivative test doesn't specify the sign of f'(c), so this option is not necessarily true.

c) f(x) has an inflection point at c: Inflection points occur when the sign of f''(x) changes. If f''(c) = 0 and the sign changes, then c is an inflection point.

d) f(x) is constant on the interval (a, b): If f''(x) = 0 on the interval (a, b), it doesn't necessarily mean that f(x) is constant. It could still have a constant slope.

Given the information, option c) seems to be the most likely, as it indicates the presence of an inflection point, which is consistent with the graph of f''(x).

Answer 2

The correct statement about the function f(x) on the interval [a, b] is: b) f(x) has a local minimum at c.

The answer is option ⇒b

Step-by-step explanation:

Based on the given information, we know that f(x) is a continuous function and its second derivative, f''(x), is shown on the interval [a, b]. Additionally, f(x) has only one critical point at c.

To determine the behavior of f(x) on the interval [a, b], we need to analyze the concavity of the function. The concavity of a function can be determined by examining the sign of its second derivative, f''(x).

1. If f''(x) > 0 for all x in the interval [a, b], it indicates that the function is concave up throughout the interval. In this case, f(x) has a local minimum at c. Therefore, option b) "f(x) has a local minimum at c" is the correct statement.

2. If f''(x) < 0 for all x in the interval [a, b], it suggests that the function is concave down throughout the interval. In this scenario, f(x) has a local maximum at c. However, this contradicts the given information that f(x) has only one critical point at c.

3. If f''(x) changes sign at c, it implies that the concavity of f(x) changes at this point. This indicates an inflection point, where the function transitions from being concave up to concave down or vice versa. However, the question specifies that f(x) has only one critical point at c, so option c) "f(x) has an inflection point at c" cannot be true.

4. If f''(x) is constant on the interval [a, b], it implies that the concavity of f(x) does not change. In this case, the function f(x) would either be concave up or concave down throughout the interval, but it would not have a critical point at c. Therefore, option d) "f(x) is constant on the interval [a, b]" is also incorrect.

To summarize, the correct statement about the function f(x) on the interval [a, b] is: b) f(x) has a local minimum at c.

The answer is option ⇒b

Your question is incomplete, but most probably the full question was:

Refer to the graph below:

The graph of f"(x), the second derivative of the continuous function f(x), is shown above on the interval a,b. On this interval, f(x) has only one critical point, which occurs at c. Which of the following statements is true about the functionf(x) on the interval a,b?

a) f(x) has a local maxiμm at c

b) f(x) has a local miniμm at c

c)f(x) has an inflection point at c

d) f(x) is constant on the interval a,b.