Question: Consider three securities with the following expected returns, standard deviations of returns, and correlations between returns:
$$
\begin{matrix}
\mu\_1=0.20, & \sigma\_1=0.31, & \rho\_{12}=\rho\_{21}=-0.2, \\
\mu\_2=0.25, & \sigma\_2=0.17, & \rho\_{23}=\rho\_{32}=0.6, \\
\mu\_3=0.17, & \sigma\_3=0.27, & \rho\_{31}=\rho\_{31}=0.15, \\
\end{matrix}
$$
Determine the weights in the minimum variance portfolio. 

My question is what is a minimum variance portfolio?  

For finding the weights I know $0.20\omega\_1+0.25\omega\_2+0.17\omega\_3=$ expected return of portfolio and $\omega\_1+\omega\_2+\omega\_3=1$ but I don't know what the expected return of the portfolio is and there are three variables but only 2 constraints?

Answer: 
Hint: Consider $q(x)=\mid p(x)\mid^2$ it is a polynomial of even degree, its derivative has an odd degree. This implies $lim\_{x\rightarrow -\infty}q'(x)=-\infty$ $lim\_{x\rightarrow +\infty}q'(x)=+\infty$. So there exists $a<0, b>0, a<b$ such that $q'(x)<0, x<a$ and $q'(x)>0, x>b$. Thus $p^2(x)$ is a decreasing function on $(-\infty,a]$ and an increasing function on $[b,+\infty)$. The minimum of the restriction of $p^2$ to $[a,b]$ exists and its square root is the minimum of $\mid p(x)\mid$.
Does $|p(x)|$ always attain $p(x)$'s minimum?

Try using a $p(x)$ such that $a\_0<0$ and $a\_0$ happens to be its minimum [example: $p(x)=x^2-1$]. Then $|p(x)|$ will never attain $a\_0$, but $|a\_0|$.

Maybe I'm misunderstanding the question. Try to word it better.