Problem source

\begin{questionparts}
\item Find the greatest and least values of $bx+a$
for $-10\leqslant x \leqslant 10$, distinguishing
carefully between the cases $b>0$, $b=0$ and $b<0$.
\item Find the greatest and least values of $cx^{2}+bx+a$,
where $c\ge0$,
for $-10\leqslant x \leqslant 10$, distinguishing
carefully between the cases that can arise
for different values of $b$ and $c$.
\end{questionparts}

Solution source

\begin{questionparts}
\item Case $b > 0$. Then $bx+a$ is increasing and the greatest value is $10b+a$, and the least value $a-10b$

Case $b=0$, then $a$ is constant and the greatest and least value is $a$

Case $b < 0$, then $bx+a$ is decreasing and the greatest value is $-10b+a$ and the least value is $10b+a$

\item If $c = 0$ we have the same cases as above.

If $ c > 0$ the consider $2cx+b$. if $b-20c > 0$ then our function is increasing on our interval and the greatest value is $100c+10b+a$ and the least value is $100c-10b+a$

If $20c+b < 0$ then our function is decreasing and that calculation is reversed.

If neither of these are true, then the minimum will be when $x = - \frac{b}{2c}$ and the max at one end point.
\end{questionparts}