The answer is $\boxed{\textbf{(E)}}$.
Quite the opposite of statement (E) is true--the sum $a+b$ is minimized when $a=b$, but it approaches $\infty$ when one of $a,b$ gets arbitrarily small.
Of the following statements, the only one that is incorrect is:
$\textbf{(A)}\ \text{An inequality will remain true after each side is increased,}$ $\text{ decreased, multiplied or divided zero excluded by the same positive quantity.}$
$\textbf{(B)}\ \text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.}$
$\textbf{(C)}\ \text{If the sum of two positive quantities is given, ther product is largest when they are equal.}$
$\textbf{(D)}\ \text{If }a\text{ and }b\text{ are positive and unequal, }\frac{1}{2}(a^{2}+b^{2})\text{ is greater than }[\frac{1}{2}(a+b)]^{2}.$
$\textbf{(E)}\ \text{If the product of two positive quantities is given, their sum is greatest when they are equal.}$