Given that the limit as h approaches 0 of (f(6 + h) - f(6))/h = -2, which of these statements must be true?
1. f'(6) exists
2. f(x) is continuous at x=6
3. f(6) < 0
that is the very definition of the derivative of f(x) at x = 6
so
it exists, it is -2
the function is contiuous because it has a unique slope at 6
All I know is that the SLOPE is negative 2 at x = 6. I do NOT KNOW if the function is +, - or 0
Isn't #2 also supposed to be true? I thought f(x) had to be continuous at the point at which we find the derivative.
I agree :)
I suppose you could have the function jump up or down at x = 6 and have the same slope of -6 on both sides of the jump, but that is a bit of a stretch.
To determine which of the statements must be true, we need to analyze the given limit expression. The limit as h approaches 0 of (f(6 + h) - f(6))/h = -2 is the definition of the derivative of f(x) at x = 6.
1. Statement 1: f'(6) exists
Since the given limit expression represents the derivative of f(x) at x = 6, we can conclude that f'(6) exists. Therefore, statement 1 must be true.
2. Statement 2: f(x) is continuous at x = 6
The limit expression gives information about the derivative of f(x) at x = 6, but it does not directly tell us about the continuity of f(x) at x = 6. Therefore, statement 2 cannot be determined solely from the given information.
3. Statement 3: f(6) < 0
The given limit expression does not provide any direct information about the value of f(6). Therefore, we cannot determine whether f(6) is less than 0 based solely on the given information.
In conclusion, statement 1 must be true, while statements 2 and 3 cannot be determined from the given information.