5. If the function $f(x)=-\frac{1}{2} x^{2}+\frac{13}{2}$ has a minimum value of $2a$ and a maximum value of $2b$ on the interval $[a, b]$, find $[a, b]$.
(2000, National High School Mathematics Competition)
(Discuss in three cases. (1) If $0 \leqslant a<b$, $f(x)$ is decreasing on $[a, b]$, so $f(a)=2 b, f(b)=2 a$. Solving gives $[a, b]=[1,3]$. (2) If $a<0<b$, by monotonicity $(f(x))_{\max }=f(0)=2 b$, which gives $b=\frac{13}{4}, a=-2-\sqrt{17}$. Thus, $[a, b]=\left[-2-\sqrt{17}, \frac{13}{4}\right]$. (3) When $a<b \leqslant 0$, by monotonicity, $a$ and $b$ do not exist. Therefore, the required intervals are $\left[-2-\sqrt{17}, \frac{13}{4}\right]$ or $[1,3]$.)