Is $-\ln(1+e^x)$ a convex function?

Question

Is $-\ln(1+e^x)$ a convex function?

My answer book says no because its second derivative is $-\dfrac{e^{2x}}{(1+e^x)^2}$ but I am sure that it is incorrect.

I have that the second derivative is $-\dfrac{e^x}{(1+e^x)^2}$.

Is the answer book right, that $-\ln(1+e^x)$ is NOT a convex function?

Answer 1

Second derivate is $-\dfrac{e^x}{(1+e^x)^2}<0$. So the correct answer is in your book: $f$ is not convex. Indeet, it is concave.