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Math Help - draw curve

  1. #1
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    Exclamation draw curve

    Sketch the grah of a continuous function f that satisfies all of the stated conditions.

    f(0) =2;f(-2) = f(2)=0; f^1(-2) = f^1(0)=f^l(2)=0;
    f^l(x)> 0 if -2 <x<0;f^1(x)<0 if x<-2 or x>0;
    f^ll(x) > 0if x<-1 or 1 <x<2;
    f^ll(x) < 0 if -1<x<1 or x>2

    Okay this problem has got me in a wirl wind of fire. Step-by-step help please.
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  2. #2
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    i don't understand how to start the problem but do you start with derivatives
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  3. #3
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    Hello, uniquereason81!

    Sketch the grah of a continuous function f(x)
    that satisfies all of the stated conditions.

    f(0) \,=\,2;\;\;f(\text{-}2) \,=\, f(2)\,=\,0\;\;{\color{blue}[1]}

    f'(\text{-}2) \,= \,f'(0)\,=\,f'(2)\:=\:0\;\;{\color{blue}[2]}

    f'(x) \;=\;\bigg\{\begin{array}{cccc}> 0 & \text{if }\text{-}2 < x <0 & {\color{blue}[3]}\\< 0 & \text{if }x<\text{-}2\:\cup\: x>0 & {\color{blue}[4]}\end{array}

    f''(x) \;=\; \bigg\{\begin{array}{cccc}> 0 & \text{if }x < \text{-}1 \:\cup\:1 < x < 2 & {\color{blue}[5]}\\ < 0 & \text{if }\text{-}1 < x < 1 \:\cup \:x > 2 & {\color{blue}[6]}\end{array}

    [1] gives us three points: . 0,2),\2,0)" alt="(-2,0),\0,2),\2,0)" />

    [2] tells that the tangents are horizontal at those three points
    Code:
                     2|
                     =o=
                      |
                      |
                      |
          - =o= - - - + - - - =o= - -
             2        |        2
                      |

    [3] tells us that the curve is uphill on [-2.0]
    [4] says it is downhill everywhere else.
    Code:
     
                      |
                      *
                    / | \
         \        /   |   \
           \    /     |     \
          - -*- - - - + - - - -*- -
            -2        |        2 \
                      |           \ 
                      |

    [5] tells us the curve is concave up to the left of -1, and on [1,2]
    Code:
                      |
        *             *
                    / |  \
         *        /   |    *
          *     *     |     *
        - - -*- -+- - + - -+- -*- -
            -2  -1    |    1   2\
                      |          \ 
                      ||

    [6] says the curve is concave down everywhere else.
    Code:
                      |
        *             *
                   *  |  *
         *       *    |    *
          *     *     |     *
        - - -*- -+- - + - -+- -*- - - -
             2  -1    |    1   2   *
                      |              * 
                      |               *
                      |

    There!

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  4. #4
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    wow thanks alot
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