QPaperGen

We have \( f(x) = 4x - \frac{1}{2}x^2 \)

\[
f'(x) = 4 - x \\
f'(x) = 0 \Rightarrow x = 4
\]

Now evaluate \( f \) at endpoints and critical point:
\[
f(-2) = 4(-2) - \frac{1}{2}(-2)^2 = -8 - 2 = -10 \]
\[f(4) = 4(4) - \frac{1}{2}(16) = 16 - 8 = 8 \]
\[f\left( \frac{9}{2} \right) = 4 \cdot \frac{9}{2} - \frac{1}{2} \cdot \left( \frac{9}{2} \right)^2 = 18 - \frac{81}{8} = 18 - 10.125 = 7.875
\]

\(\textbf{Absolute maximum}\) is \( 8 \) at \( x = 4 \) and
\(\textbf{Absolute minimum}\) is \( -10 \) at \( x = -2 \)