# draw curve

• Apr 23rd 2008, 04:33 PM
uniquereason81
draw curve
(Doh)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.
• Apr 23rd 2008, 09:18 PM
uniquereason81
i don't understand how to start the problem but do you start with derivatives
• Apr 23rd 2008, 09:25 PM
Soroban
Hello, uniquereason81!

Quote:

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

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

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

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

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

[1] gives us three points: .$\displaystyle (-2,0),\:(0,2),\:(2,0)$

[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!

• Apr 23rd 2008, 09:39 PM
uniquereason81
wow thanks alot