Lời giải:

• \( f(x) = \begin{cases} x^2 - x - 6, & x \leq -1 \vee x \geq 5 \\ -x^2 + 7x + 4, & -1 < x < 5 \end{cases} \)
• Đạo hàm: \( f'(x) = \begin{cases} 2x - 1, \text{ nếu } x < -1 \vee x > 5 \\ -2x + 7,  \text{ nếu }  -1 < x < 5 \end{cases} \)
• \( \max\limits_{[-2,6]} f(x) = \max \{ f(-2), f(6), f(-1), f(5), f\left(\frac{1}{2}\right), f\left(\frac{7}{2}\right) \} \) \( = \max \left\{ 0, 24, -4, 14, \frac{29}{4}, \frac{65}{4} \right\} = 24 \)