Determine if each function is differentiable at x=1. If it is find the dirivative, if not explain y not.

a) f(x) = { 3x-2 if x<1 and x^3 if x >=1

b) f(x) = { 2x+1 if x<1 and x^2 if x>=1

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- Apr 22nd 2007, 12:10 PMlearn18Differentiable
Determine if each function is differentiable at x=1. If it is find the dirivative, if not explain y not.

a) f(x) = { 3x-2 if x<1 and x^3 if x >=1

b) f(x) = { 2x+1 if x<1 and x^2 if x>=1 - Apr 22nd 2007, 12:15 PMJhevon
We are differentiable as long as we are continuous. so we know f(x) is differentiable at x = 1 if it is continuous at x = 1. you can show this by finding the left and right hand limits. i will just show you the graph, so you can see it's continuos.

as you see, f(x) is differentiable here.

using f(x) = x^3

=> f ' (x) = 3x^2 - Apr 22nd 2007, 12:17 PMCaptainBlack
Because these functions are both piecewise differentiable all we need to

show is that:

lim_{x->1-} f(x) = lim_{x->1+} f(x),

and that:

lim_{x->1-} f'(x) = lim_{x->1+} f'(x),

If both of these hold then the function is differentiable at x=1, otherwise they

are not.

For a) lim_{x->1-} f(x) = 1, lim_{x->1+} f(x) = 1, and lim_{x->1-} f'(x) = 3,

lim_{x->1+} f'(x) = 3, so f is differentiable at x=1.

For b) lim_{x->1-} f(x) = 3, lim_{x->1+} f(x) = 1, so f is not differentiable

at x=1.

RonL - Apr 22nd 2007, 12:18 PMJhevon
- Apr 22nd 2007, 12:21 PMJhevon
I was not aware that we have to check the left and right hand limits for f', i thought once it checked out for f we can assume it would check out for f'. is this an incorrect assumption? can you give a counter example, where f is continous at the point but f' isn't

EDIT: nevermind, learn18 provided us with a counter example in a subsequent problem, see http://www.mathhelpforum.com/math-he...erntiable.html