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Math Help - Show that f is not differentiable at 0

  1. #1
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    Show that f is not differentiable at 0

    Hey guys, I need help with this question that has been bugging me...
    Let f(x) = x3/5. Show that f is not differentiable at zero. Show that y = x3/5 has a vertical tangent line at (0,0)

    I don't know how to prove the statements... Any tips and pointers (plus the method) will help alot...

    Thanks!
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  2. #2
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    Re: Show that f is not differentiable at 0

    There are several different ways to do this, depending on what you can or want to use.

    Of course, a function of one variable is "differentiable" at x= a if and only if \lim_{x\to h} \frac{f(a+h)- f(a)}{h} exists. It is NOT differentiable if that limit does not exist. So you could show that \lim_{h\to 0}\frac{h^{3/5}- 0}{h} does not exist.

    Or you could use the fact that, while the derivative of a function is not necessarily continuous, it does have the "intermediate value property". In particular, if the derivative at x= a exists, then \lim_{x\to a}f'(x)= a. So what is the derivative of f(x)= x^{3/5}? What is the limit of that derivative as x goes to 0?
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