1

Give an example of a function $f$ that is differentiable at $x=a$ such that $f'(a)\ne0$ but yet $f$ attains a relative extremum at $x=a$.

By definition, the best I've gotten so far is using a function like $f(x)=x$ for $x=[0,1]$ since the maximum on that domain would be 1, and $f'(1)=1$, but I'm not sure if that's air tight (or even correct).

user65384
  • 614
  • 9
    I don't think there's such a thing: to be differentiable at $,x=a,$ requires the function being defined in some open non-empty neighborhood of that point, and thus for the point to be an extremum one the derivative must be zero. One sided stuff isn't appliable here, imo. – DonAntonio Mar 14 '13 at 18:33
  • 2
    Looks fine, and is probably the simplest case you'll be able to find. – vonbrand Mar 14 '13 at 18:33
  • @user65384 You should check your definition of derivative. Is it defined only on interior points? – Git Gud Mar 14 '13 at 18:55

1 Answers1

1

You know that if $f$ attains a maximum or minimum in a point internal to the domain where the function is differentiable then $f'=0$ in the point (Fermat's theorem). Hence the only possibility is the the point is on the boundary of the domain. Your example is hence correct.

Emanuele Paolini
  • 21,447