Question

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Answer 1

If f'(3) exists then the function is differentiable. Every differentiable function is continuous, meaning the limit of f(x) as x approaches 3 is equal to f(3)

2. The first derivative test tests intervals between critical points, where f'(x) = 0. If the sign of f'(x) changes at a critical point, then you know the slope (f'(x)) of the graph changes from positive to negative or negative to positive, meaning there is a local minimum or maximum at that critical point.

Answer 2

I'm not exactly sure what you mean.

Answer 3

If the function has no critical points, then the function must always be increasing or decreasing. Linear functions are probably the best example of this.

Answer 4

Ok awesome that was my question. Thank you so much!

Answer 5

You're welcome