relative max/mins: how many real zeroes does poly have?

[degreeplus]

A polynomial p(x) has a relative maximum at (-2,4), a relative minimum at (1,1), a relative maximum at (5,7), and no other critical points. How many real zeros does p(x) have?

(A) One (B) Two (C) Three (D) Four (E) Five

Here i would think that since there are only 3 critical points given and no other information other than that those are the only critical points that p(x) would not have any real zeros.

I'm confused what they mean by "real zeros" because from what I think it sounds like they're asking how many times does p(x) cross the x-axis and I can't see how p(x) crosses the x-axis at all if it goes from a rel. max of (-2,4) to a rel. min of (1,1) then to a rel. max of (5,7) and that is all the critical values. Thanks for any help.

Also the answer is (B).

 

[soroban]

Hello, degreeplus!

A polynomial p(x) has a relative maximum at (-2,4), a relative minimum at (1,1),
a relative maximum at (5,7), and no other critical points.
How many real zeros does p(x) have?

. . (A) One . . (B) Two . . (C) Three . . (D) Four . . (E) Five

Did you make a sketch?

               |                (5,7)
               |                  *
               |               *     *
      (-2,4)   |              *
         *     |                      *
       *    *  |             *
      *       *|
               *            *
     *         |*          *
               |  *      *
               |     *
               |   (1,1)
    - - - - - -+ - - - - - - - - - - - -
               |

If there are no more critical points, we can make some assumptions:
. . that the graph continues to decrease at the far left and the far right.

Hence, there will be two x-intercepts.