A Negative Sliver

Let $a$ be a constant and $f(x) = x\ln x + ax^2$.

1. When $a = 1$, find the equation of the tangent line to $f(x)$ at $x = 1$. 2. Determine the number of zeros of $f(x)$, discussing all cases for $a$. 3. If $f(x)$ has two extreme points $x_1 < x_2$, prove that