Re: help with a basic differentiability and derivative function

David L. Wilson wrote:
"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.


Right. Since the second expression holds up to x = 1 the derivative is defined by it at x = 1. The first expression only holds for x > 1 so the derivative is different for x > 1, not at x = 1.
.