If \(f\) is continuous for \(a\leq x \leq b\) and differentiable for \(a\lt x\lt b\), which of the following could be false?

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There is no requirement that the derivative be 0 for a continuous and differentiable function. For example, here is a simple function that is both continuous and differentiable but excludes any points where the derivative is 0:

$$ f(x) = x $$ $$ f'(x) = 1 \tag*{\scriptsize at all points}$$

The first choice is a restatement of the mean value theorem, which is true for all differentiable functions.

All functions have a minimum value provided at least one point exists.

If a function is continuous, the area under the curve exists and we should be able to integrate the function.