The graph of f ′(x) is continuous and increasing with an x-intercept at x = 0. Which of the following statements is false?
The graph of f is always concave up.
The graph of f has an inflection point at x = 0.
The graph of f has a relative minimum at x = 0.
The graph of the second derivative is always positive.
What does this mean steve just pls put the answer i dont know why you always do this steve I just want the answer dude if you can include reasoning that really helps but just reasoning without an answer? I am going to cry :(. This is just cryptic not helpful.
Steve be bad
literally crying at "please just put the answer steve"
To find the false statement, let's analyze each option one by one:
1. "The graph of f is always concave up."
To determine the concavity of the function f, we need to examine the graph of the second derivative, f''(x). If f''(x) is positive for all x, then the graph of f is indeed always concave up. So, we should investigate the truthfulness of the next statement before concluding on this option.
2. "The graph of f has an inflection point at x = 0."
To check whether the graph of f has an inflection point at x = 0, we need to examine the concavity changes of f'. If f' switches from increasing to decreasing (or vice versa) around x = 0, then there is an inflection point at x = 0.
3. "The graph of f has a relative minimum at x = 0."
For f to have a relative minimum at x = 0, f' needs to change from positive to negative at that point. This would indicate a switch from increasing to decreasing.
4. "The graph of the second derivative is always positive."
This statement implies that f''(x) > 0 for all x. If this is true, then the graph of f is always concave up, and therefore, it cannot have any inflection points or relative minima.
Now, let's consider each option:
1. "The graph of f is always concave up."
We are not certain if this statement is true yet, so we cannot classify it as false.
2. "The graph of f has an inflection point at x = 0."
If f' is always continuous and increasing, then it cannot change concavity. Therefore, f cannot have an inflection point at x = 0. This statement is false.
3. "The graph of f has a relative minimum at x = 0."
Since the x-intercept of f' is at x = 0, and it is increasing, f cannot have a relative minimum at x = 0. This statement is also false.
4. "The graph of the second derivative is always positive."
Given that f' is continuous and increasing, this option is true.
In conclusion, the false statement is:
"The graph of f has an inflection point at x = 0."
consider the case
f'(x) = x