Help me check my calculus
1. The graph of f '(x) is continuous, positive, and has a relative maximum at x = 0. Which of the following statements must be true?
a. The graph of f is always concave down.
b.The graph of f is always increasing.
c. The graph of f has a relative maximum at x = 0.
d. The graph of f has a relative minimum at x = 0.
I chose B. Is it correct?
Yes, the slope is always positive, goes up forever.
To check if your answer is correct, let's go through each statement and determine if it must be true based on the given information.
a. The graph of f is always concave down.
To determine the concavity of f, we need to look at the second derivative, f''(x). Since no information about f''(x) is provided, we cannot conclude that the graph of f is always concave down based solely on the given information. Therefore, statement (a) cannot be assumed to be true.
b. The graph of f is always increasing.
Given that f'(x) is continuous and positive, it means that the derivative function is always positive. This indicates that the original function, f(x), is always increasing. Therefore, statement (b) must be true based on the given information.
c. The graph of f has a relative maximum at x = 0.
This statement is explicitly mentioned in the question. It states that f'(x) has a relative maximum at x = 0. Since f'(x) represents the derivative of f(x), this means that the graph of f has a relative maximum at x = 0. Therefore, statement (c) can be assumed to be true.
d. The graph of f has a relative minimum at x = 0.
The given information does not provide any details about the behavior of f(x) near x = 0, apart from the fact that its derivative, f'(x), has a relative maximum at x = 0. Therefore, we cannot conclude whether the graph of f has a relative minimum at x = 0 based on the given information. Thus, statement (d) cannot be assumed to be true.
Considering our analysis, your choice of B, "The graph of f is always increasing," is indeed correct based on the given information.
Therefore, from the given options, statement B is the one that must be true.