How do you prove that a function is/isn't differentiable at its endpoints?

Question

anonymous · Answer

The question that I am doing only requires identifying it graphically

anonymous · Answer

But I would also like to know how to do it with limits, or however they expect me to do it in calculus

precal · Answer

I would use the Theorem about Differentiability.
If the function is differentiable then the function is continuous.

anonymous · Answer

Do you mean the other way? If a function is continuous then it is differentiable?

precal · Answer

But you can't use the converse of it
ie If a function is continuous then it is differentiable.  This is not a true statement.

anonymous · Answer

My issue is that i don't know if the function is differentiable. I am supposed to identify which points the function is differentiable on in a closed interval

precal · Answer

attach the graph

precal · Answer

draw slopes on the graph.  If you can draw the slopes then that is your derivative

anonymous · Answer

|dw:1410623917120:dw|

anonymous · Answer

The function is continuous at its endpoints

precal · Answer

Then this particular graph is differentiable but the endpoints look like they have vertical  tangents

anonymous · Answer

They aren't. They don't look that extreme in the book. My question is how do you know that it is differentiable for its domain?

precal · Answer

maybe vertical asymptotes

precal · Answer

Usually the endpoints are not included in the theorems about differentiability

anonymous · Answer

Thats a problem

precal · Answer

if the endpoints are not included, it is differentiable

anonymous · Answer

The endpoints are included in the problem

precal · Answer

Are you looking at a particular theorem?

anonymous · Answer

However I am supposed to solve the rpobk

anonymous · Answer

*problem

precal · Answer

@dumbcow

precal · Answer

let's see if dumbcow can help us

anonymous · Answer

Ok, thanks

dumbcow · Answer

hmm well as long as its continuous and smooth the function is differentiable at all points 

if there is a sharp curve where its not smooth like an absolute value, then its not differentiable

anonymous · Answer

What if the endpoints aren't continuous?

dumbcow · Answer

is the function defined at the endpoint?

anonymous · Answer

On this particular problem, it is continuous.

dumbcow · Answer

for example, say function is x^2 for -1<=x<=2

i know that derivative is 2x so the slope of tangent at endpoints are -2 and 4
thus since the endpoints are defined and lie along a smooth curve, the endpoints are differentiable

anonymous · Answer

They don't give us the function, they only give us the domain