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Differentiability

I have a function which is f(x)=x+|x|^3

I need to determine whether f is differentiable at 0 and rigorously justify my answer but im struggling so can someone help?
Reply 1
Avatar for mqb2766
mqb2766
21
From first principles, consider the derivative(s) f'(0) when
* h is positive
* h is negative
Are the two values equal, if not ...

Original post by Pepsey
I have a function which is f(x)=x+|x|^3

I need to determine whether f is differentiable at 0 and rigorously justify my answer but im struggling so can someone help?
Original post by mqb2766
From first principles, consider the derivative(s) f'(0) when
* h is positive
* h is negative
Are the two values equal, if not ...

f'(0)=1 so what do I do from there?
I've never seen the h is positive part in my life tbh
Reply 3
Avatar for mqb2766
mqb2766
21
The derivative at zero is defined by
(f(0+h) - f(0)))/h
and considering what this tends to as h goes to 0.
Evaluate this when h is positive and when h is negative. If the derivatives exist and are both equal, you're good. If not, ...
Original post by Pepsey
f'(0)=1 so what do I do from there?
I've never seen the h is positive part in my life tbh
Original post by Pepsey
f'(0)=1 so what do I do from there?
I've never seen the h is positive part in my life tbh

Note that if you know that f′(x0)=df'(x_0) = d, then by definition, you know that f is differentiable at x0x_0 with derivative d.

So I am forced to ask how you know that f'(0) = 1?

What definition of differentiable have you been given in lectures?
Reply 5
Avatar for Ryanzmw
Ryanzmw
16
Original post by Pepsey
I have a function which is f(x)=x+|x|^3

I need to determine whether f is differentiable at 0 and rigorously justify my answer but im struggling so can someone help?

Just looking at it, do you think the left and right hand limits are going to agree?

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