1. ## Graphing Help???

I am supposed to a graph the function f(x) given the following characteristics:

• f(0)= 4 and f(6)= 0
• f '(x)< 0 if x< 2 and x> 4
• f '(2) does not exist and f '(4)=0
• f '(x)>0 if 2< x< 4
• f ''(x)< 0, x ¹ 2
I understand the first one gives the points (0,4) and (6,0) but am confused with the other four. Please Help

2. Hello, mathkid2007!

Graph the function $f(x)$ given the following characteristics:

$(1)\;f(0)= 4\text{ and }f(6)= 0$
$(2)\;f '(x)< 0 \text{ if }x< 2\text{ and }x> 4$
$(3)\;f '(2)\text{ does not exist and }f '(4)=0$
$(4)\;f '(x)>0\text{ if }2< x< 4$
$(5)\;f ''(x)< 0,\;x \neq 2$

(1) gives us two points: (0,4) and (6,0).
Code:
|
4*
|
|
|
- - + - - - - - - - - - - - * - - -
|                       6
|

(2) says the slope is negative for $x < 2$ and $x > 4.$
Code:
|
|
*   |
4*
|  *              *
|    *               *
|     *                * 6
- - + - - - - + - - - + - - * - - -
|         2       4
|                         *
|

(3) says there is a vertical asymptote at $x = 2$
. . and a horizontal tangent at $x = 4.$
Code:
|      x=2
*   |       :
4*       :
|  *    :     --*--
|    *  :       :   *
|     * :       :      * 6
- - + - - - + - - - + - - - * - - -
|      *:       4
|       :                *
|      *:
|       :

(4) says the slope is positive on (2,4).
(5) says the graph is always concave down.
Code:
|      x=2
|       :
*   |       :
4*       :
|  *    :       *
|    *  :   *   :   *
|     * :  *    :      * 6
- - + - - - + * - - + - - - * - - -
|      *:2      4
|       :*               *
|      *:
|       :

. . . There!