sol-2.11.tex

An interval can have either both extrema $<0$, or both extrema $\geq 0$, or one extrema $<0$ and the other $\geq 0$. Since we multiply two intervals, in total there are $3^2$ cases.  Eight of these cases require two multiplications only, while the remaining one requires four multiplications.
\begtt\scm
(define (mul-interval x y)
  (let ((a (lower-bound x))
        (b (upper-bound x))
        (c (lower-bound y))
        (d (upper-bound y)))
    (cond ((and (>= a 0) (>= c 0))
           (make-interval (* a c) (* b d)))
          ((and (>= a 0) (< c 0) (>= d 0))
           (make-interval (* b c) (* b d)))
          ((and (>= a 0) (< d 0))
           (make-interval (* b c) (* a d)))
          ((and (< a 0) (>= b 0) (>= c 0))
           (make-interval (* a d) (* b d)))
          ((and (< a 0) (>= b 0) (< c 0) (>= d 0))
           (make-interval (min (* a d) (* b c))
                          (max (* b d) (* a c))))
          ((and (< a 0) (>= b 0) (< d 0))
           (make-interval (* b c) (* a c)))
          ((and (< b 0) (>= c 0))
           (make-interval (* a d) (* b c)))
          ((and (< b 0) (< c 0) (>= d 0))
           (make-interval (* a d) (* a c)))
          ((and (< b 0) (< d 0))
           (make-interval (* b d) (* a c))))))
\endtt

We can also avoid redundant checks by rewriting the procedure in the following way:
\begtt\scm
(define (mul-interval x y)
  (let ((a (lower-bound x))
        (b (upper-bound x))
        (c (lower-bound y))
        (d (upper-bound y)))
    (if (>= a 0)
        (if (>= c 0)
            (make-interval (* a c) (* b d))
            (if (>= d 0)
                (make-interval (* b c) (* b d))
                (make-interval (* b c) (* a d))))
        (if (>= b 0)
            (if (>= c 0)
                (make-interval (* a d) (* b d))
                (if (>= d 0)
                    (make-interval (min (* a d) (* b c))
                                   (max (* b d) (* a c)))
                    (make-interval (* b c) (* a c))))
            (if (>= c 0)
                (make-interval (* a d) (* b c))
                (if (>= d 0)
                    (make-interval (* a d) (* a c))
                    (make-interval (* b d) (* a c))))))))
\endtt