$\begin{array}{l}
f){x^2} + xy + {y^2} + 5\\
= {x^2} + 2.\frac{1}{2}xy + \frac{1}{4}{y^2} + \frac{3}{4}{y^2} + 5\\
= {\left( {x + \frac{1}{2}y} \right)^2} + \frac{3}{4}{y^2} + 5 \ge 5\\
\Rightarrow \min F = 5 \Rightarrow \left\{ \begin{array}{l}
y = 0\\
x = 0
\end{array} \right.\\
g)G = \left( {x - 3} \right)\left( {x + 5} \right) + 41\\
= {x^2} + 2x - 15 + 41 = {x^2} + 2x + 26 = \left( {{x^2} + 2x + 1} \right) + 25\\
= {\left( {x + 1} \right)^2} + 25 \ge 25 \Rightarrow \min G = 25\\
\Rightarrow x + 1 = 0 \Rightarrow x = - 1\\
h)H = \left( {x - 2} \right)\left( {x + 4} \right) - 11\\
= {x^2} + 2x - 8 - 11 = {x^2} + 2x + 1 - 9 - 11\\
= {\left( {x + 1} \right)^2} - 20 \ge - 20 \Rightarrow \min H = - 20 \Rightarrow x + 1 = 0\\
\Rightarrow x = - 1
\end{array}$
