I don't understand why \( -log(x \) is not smooth, as is twice differentiable on \( \mathbb{R}^+ \) with \( f''(x) = \frac{1}{x^2} > 0 \forall x \in \mathbb{R}^+ \) Using theorem 1.9 along with lemma 6.1 (easy direction of lemma 2.4) shouldnt we get that f is smooth?

Using lemma 6.1, I think the second derivative needs to be bounded to ensure smoothness. However it is not the case here, because for x close to zero the second derivative goes to infinity. Hope it helps!

## Q5 exam 2020

Hello,

I don't understand why \( -log(x \) is not smooth, as is twice differentiable on \( \mathbb{R}^+ \) with \( f''(x) = \frac{1}{x^2} > 0 \forall x \in \mathbb{R}^+ \) Using theorem 1.9 along with lemma 6.1 (easy direction of lemma 2.4) shouldnt we get that f is smooth?

Thank you for any help!

Cheers

Yann

Using lemma 6.1, I think the second derivative needs to be bounded to ensure smoothness. However it is not the case here, because for x close to zero the second derivative goes to infinity. Hope it helps!

Oh yeah this was very silly, I was only checking for x tending to infinuty.. thank you for the help!

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