Re: help with a basic differentiability and derivative function


"David Wilkinson" <david@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message
news:do8f40$pjb$1@xxxxxxxxxxxxxxxxxxxxxxx
> disanalysis wrote:
>> very basic problem...can anyone help me with the proof?
>>
>> f(x) = x^2-3x+8 if x>1
>> f(x) = 2x^2+x+3 if x=<1
>>
>> using epsilon-delta property can you prove that f(x) is differentiable
>> at all points except x=1
>> and that it is not differentiable at 1?
>>
>> thanx
>>
> At x = 1 f(x) = 6 from both expressions.
>
> for x > 1, df(x)/dx = 2x-3 -> -1 as x -> 1 from the first expression, and
>
> For x <= 1, df(x)/dx = 4x+1 = 5 at x=1.
>
> So the first derivative of f(x) is discontinuous at just above x = 1. But,
> at x = 1 it is differentiable from the second expression which holds up to
> and including x = 1.

Wrong. The derivative does NOT exit at x=1 because the limit definining the
derivative at that point does not exist as the limit defining the derivative
at x=1 from the left is not equal to the limit defining the derivative at
x=1 from the right. Or were you intentionally giving him a wrong answer on
this obvious homework problem.

Graphically, there is a sharp turn at x=1, hence, the derivative does not
exist at the point (1, 6) although the function is continuous there.


.