How many of the following are equal to $x^x+x^x$ for all $x>0$?
$\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textbf{IV:}\ (2x)^{2x}$
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
We look at each statement individually.
$\textbf{I:}\ 2x^x$. We note that $x^x+x^x=x^x(1+1)=2x^x$. So statement $\textbf{I}$ is true.
$\textbf{II:}\ x^{2x}$. We find a counter example which is $x=1$. $2\neq 1$. So statement $\textbf{II}$ is false.
$\textbf{III:}\ (2x)^x$. We see that this statement is equal to $2^xx^x$. $x=2$ is a counter example. $8\neq 16$. So statement $\textbf{III}$ is false.
$\textbf{IV:}\ (2x)^{2x}$. We see that $x=1$ is again a counter example. $2\neq 4$. So statement $\textbf{IV}$ is false.
Therefore, our answer is $\boxed{\textbf{(B) }1}$.
--Solution by [TheMaskedMagician](https://artofproblemsolving.comhttp://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=200685)