If a function f is continuous for all x and if f has a relative minimum at (-1,4) and a relative minimum at (-3,2), which of the following statements must be true?

(a) The graph of f has a point of inflection somewhere between x = -1 and x=3
(b) f'(-1) = 0
(c) The graph of f has a horizontal asymptote
(d) The graph of f has a horizontal tangent line at x=3
(e) The graph of f intersects both axes

If it has a minimum at x = -1 then the slope, f'(-1) is indeed zero there.

To determine which of the statements must be true, let's consider the given information about the function f.

Given:
- f is continuous for all x.
- f has a relative minimum at (-1, 4) and a relative minimum at (-3, 2).

Let's analyze each statement:

(a) The graph of f has a point of inflection somewhere between x = -1 and x = 3.

Based on the information given, we cannot conclude whether there is a point of inflection between x = -1 and x = 3. The presence of relative minima does not guarantee the existence of a point of inflection. Therefore, statement (a) may or may not be true.

(b) f'(-1) = 0.

Since the function f has a relative minimum at (-1, 4), the derivative of f must be zero at x = -1. Therefore, statement (b) is true.

(c) The graph of f has a horizontal asymptote.

The given information does not provide any details about the behavior of f at large x-values. Therefore, we cannot determine whether the graph of f has a horizontal asymptote or not. Statement (c) may or may not be true.

(d) The graph of f has a horizontal tangent line at x = 3.

The given information only specifies the relative minima at (-1, 4) and (-3, 2), but it does not give any information about the behavior of f at x = 3. Therefore, we cannot determine whether the graph has a horizontal tangent line at x = 3. Statement (d) may or may not be true.

(e) The graph of f intersects both axes.

Since the relative minima of the function occur at (-1, 4) and (-3, 2), we know that the graph of f intersects the y-axis since the y-values of these minima are not equal to zero (f(-1) = 4 and f(-3) = 2). However, we cannot determine whether the graph intersects the x-axis based on the given information about the relative minima. Therefore, statement (e) may or may not be true.

In summary:
Statement (b) is true: f'(-1) = 0.
Statements (a), (c), (d), and (e) may or may not be true - we cannot determine their validity based on the given information.

To determine which of the statements must be true, we need to analyze the given information about the function f and understand the concepts involved.

1. A relative minimum: A relative minimum occurs at a point where the function is lower than surrounding points, indicating a "valley" in the graph of the function.

2. Continuous function: A continuous function implies that there are no gaps or jumps in the graph. It can be drawn without lifting the pen from the paper.

Based on this information, we can analyze each statement:

(a) The graph of f has a point of inflection somewhere between x = -1 and x = 3:
To determine if this statement is true, we need additional information about the function. Relative minima do not guarantee the presence of a point of inflection. Therefore, this statement cannot be concluded based on the given information.

(b) f'(-1) = 0:
To test this statement, we need to analyze the derivative of f. If f has a relative minimum at x = -1, the derivative of f should be zero at that point. So, this statement is likely to be true. Take the derivative of f and evaluate it at x = -1 to confirm.

(c) The graph of f has a horizontal asymptote:
The presence of relative minima does not give direct information about the existence of a horizontal asymptote. Therefore, this statement cannot be concluded based on the given information.

(d) The graph of f has a horizontal tangent line at x = 3:
The presence of relative minima at x = -1 and x = -3 does not provide direct information about the tangent line at x = 3. Therefore, this statement cannot be concluded based on the given information.

(e) The graph of f intersects both axes:
Since f has a relative minimum at (-1,4) and another relative minimum at (-3,2), it implies that the function's graph does not intersect the x-axis (since the function has a minimum value above it). However, it does not provide direct information about the y-axis. Therefore, this statement cannot be concluded based on the given information.

In summary, based on the given information, statement (b) is the only one that must be true: f'(-1) = 0. The other statements cannot be concluded without further information about the function f.