the c(t) equation, as you can see, is a quadratic equation, and thus its graph is a parabola, the leading term's coefficient is positive, so the parabola looks like the picture below
and the lowest point is at the vertex
c(t) reaches it's lowest point there
[tex]\bf \textit{vertex of a parabola}\\ \quad \\
\begin{array}{lllccllll}
c(t)&=&1t^2&-10t&+76\\
&&\uparrow &\uparrow &\uparrow \\
&&a&b&c
\end{array} \qquad
\left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)
\\\\\\\\
\textit{so, the cost } \boxed{{{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}}\textit{ will be the lowest at year }\boxed{-\cfrac{{{ b}}}{2{{ a}}}}[/tex]
