budafeet57 said:b, d are right because f'(c) is differentiable over the interval
What the claim is saying is that there exists a well-defined maximum value for f. How would the function f need to behave for this claim not to be true?budafeet57 said:I am not sure about e. Can anyone explain to me?
Ah I see, b is the answer. Because f is not a constan, so it can only be linear. When it's differentiated it can't equal to zero under [-2,1].clamtrox said:Are you sure about that? What about a linear function? Then f'(x) = constant ≠ 0.What the claim is saying is that there exists a well-defined maximum value for f. How would the function f need to behave for this claim not to be true?
budafeet57 said:Ah I see, b is the answer. Because f is not a constan, so it can only be linear. When it's differentiated it can't equal to zero under [-2,1].
clamtrox said:That makes no sense at all. Why won't you draw a picture to get some idea of what's happening, if you want to reason like this without any mathematical proofs.
A function is continuous over a closed interval if it is defined and has no breaks or gaps in its graph over that interval. This means that the function can be drawn without lifting the pen from the paper.
A function is continuous over a closed interval if it satisfies the three conditions of continuity: 1) the function is defined at every point in the interval, 2) the limit of the function as x approaches a point within the interval exists, and 3) the value of the function at that point is equal to the limit. If any of these conditions are not met, the function is not continuous over the closed interval.
Continuity refers to the smoothness of a function over an interval, while differentiability refers to the existence of a derivative at every point in that interval. A function can be continuous but not differentiable at certain points, and vice versa.
Yes, a function can be continuous but not differentiable at certain points within a closed interval. This usually occurs when there is a sharp turn or corner in the graph of the function, as the derivative does not exist at these points.
In order to prove that a function is differentiable over a closed interval, we must show that the derivative exists at every point within the interval. This can be done by using the definition of the derivative, taking the limit as h approaches 0 of the difference quotient, and showing that the limit exists and is the same for every point in the interval.