Maximum and minimum value question

[afcwestwarrior]

what does differentiable at 2 mean

what does this mean , my question says sketch the graph of a function who has a local maximum at 2 and is differentiable at 2,

what does it mean by it is differentiable at 2,

 

[afcwestwarrior]

does this mean that the slope is 0 at 2

 

[afcwestwarrior]

well if u need to know what this means, it means if it is differentiable at 2 it is continuous at 2

 

[afcwestwarrior]

one more thing how do i sketch this graph, all it gives me is it is continuous at 2, does it matter how i sketch this graph, does it have to be a certain type, does it have to look a certain way, in the back of my book it is a parabola and its continuous on the negative side

 

[baolkvn]

Well, differentiable and continuous is not equivalent. continuous if differentiable, but if continous, we can't conlude it is differentiable.
2 formulas below are definition of continuous and differentiable properties of a funtion, for example,F(x) :
+ F(x) is continuous at x0 <=> limit of F(x) when x->x0 is equal to F(x0)
+ F(x) is differentiable at x0 <=> limit of [F(x)-F(x0)]/[x-x0] when x->x0 exists (that value is so called F'(x0) )
Anyway, note that : "differentiable" and "continous" is not equivalent. "continous" if differentiable, but if "continous", we can't conlude it is differentiable.
If you have anymore question, feel confidently to ask me.

 

[Mystic998]

Well, strictly speaking, it doesn't have to look particularly normal to satisfy the requirements. But likely for your purposes, you're going to want something that's continuous in an interval around 2 and appears "smooth" at 2 (i.e., it has no sharp edge).

 

[Gib Z]

does this mean that the slope is 0 at 2

Yes. It also means it is continuous at 2. Seeing as they had a parabola, it seems this function is one where the domain is limited.