The graph of f '(x) is continuous and decreasing with an x-intercept at x = -3. Which of the following statements must be true?

To answer this question, we need to understand the properties of the derivative function f '(x) and how they relate to the original function f(x).

First, let's examine the given information:
1. The graph of f '(x) is continuous: This means that f '(x) has no breaks, holes, or jumps in its graph.
2. The graph of f '(x) is decreasing: This means that as x increases, the values of f '(x) are getting smaller. In other words, the slope of f(x) is decreasing.
3. The x-intercept of f '(x) is at x = -3: This means that f '(x) crosses the x-axis at x = -3, which means that f(x) has a critical point or an extremum at x = -3.

Now, let's determine which statements must be true based on this information:

1. f(x) must be continuous: Since f '(x) is continuous, it implies that f(x) is differentiable and continuous as well. Therefore, this statement must be true.

2. f(x) must be decreasing: Since f '(x) is decreasing, it implies that the slope of f(x) is decreasing. This means that f(x) is getting flatter as x increases. So, this statement must be true.

3. f(x) must have a local maximum at x = -3: Since the x-intercept of f '(x) is at x = -3, it indicates that f(x) has a critical point or an extremum at x = -3. However, the exact nature of this extremum cannot be determined solely based on the given information. So, this statement is not necessarily true.

In conclusion, the statements that must be true are:

1. f(x) must be continuous.
2. f(x) must be decreasing.

The statement about a local maximum at x = -3 cannot be determined with certainty based solely on the given information.

When f'(x) > 0, the function is increasing.

Conversely, f'(x) <0, function is decreasing.

Thus the function is increasing in the interval (-∞,-3], and decreasing on [-3,∞). This makes a maximum at x=-3.

Since we do not know anything about f"(x), we do not know about the concavity of the function.

Since we are given f'(x) is a decreasing function, its derivative must be negative. Hence f ''(x) < 0, and f is concave down for all x.

all of them?

f is concave down with a max at x = -3