drawing a curve based on derivatives of piecewise function

SilentSymphony

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Jun 30, 2007
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okay... the problem is as stated:

Sketch a smooth connected curve y=f(x) with

f(-2) = 8, f'(2) = f' (-2) = 0

f(0) = 4, f'(x)<0 for |x| <2

f(2) = 0, f''(x)<0 for x < 0

f'(x) > 0 for |x| > 2, f''(x) > 0 for x>0


im not asking for the answer, i just would like to know how to start working on this problem... any guidance yall can give would be awesome! :D
 
Re: drawing a curve based on derivatives of piecewise functi

SilentSymphony said:
Sketch a smooth connected curve y=f(x) with
f(-2) = 8, f'(2) = f' (-2) = 0
f(0) = 4, f'(x)<0 for |x| <2
f(2) = 0, f''(x)<0 for x < 0
f'(x) > 0 for |x| > 2, f''(x) > 0 for x>0
The given tells us that the graph is decreasing on [-2,2]. It is increasing elsewhere.
It has a max at (-2,8).
It is concave down for x<0 and up elsewhere.
 
Re: drawing a curve based on derivatives of piecewise functi

Hello, SilentSymphony!

We are given a lot of information about the curve . . .


Sketch a smooth connected curve y=f(x)\displaystyle y\:=\:f(x) with:

f(2)=8,  f(0)=4,  f(2)=0\displaystyle f(-2)\,=\,8,\;f(0)\,=\,4,\;f(2)\,=\,0

Plot these points: \(\displaystyle \:(-2,8),\:(0,4),\:(2,0)\)


f(2)=f(2)=0\displaystyle f'(2)\,=\,f'(-2)\,=\,0

There are horizontal tangents at: (2,8)\displaystyle \,(-2,8) and (2,0)\displaystyle (2,0)


f(x)<0 for x<2\displaystyle f'(x)\,<\,0\text{ for }|x|\, <\,2

On the interval (2,2)\displaystyle (-2,\,2), the graph decreasing.


f(x)>0 for x>2\displaystyle f'(x)\,>\,0\text{ for }|x|\,>\,2

On the intervals (,2)\displaystyle (-\infty,\,-2) and (2,)\displaystyle (2,\,\infty), the graphing is increasing.


f(x)<0 for x<0\displaystyle f''(x)\,<\,0\text{ for }x\, <\, 0

For x<0\displaystyle x\,<\,0, the graph is concave down: \displaystyle \cap


f(x)>0 for x>0\displaystyle f''(x)\, >\, 0\text{ for }x\,>\,0

For x>0\displaystyle x\,>\,0, the graph is concave up: \displaystyle \cup


The graph must look something like this:
Code:
           (-2,8)     |
              *       |
          *       *   |
        *           * |                *
       *             *|
                      |
      *               *(0,4)          *
                      |
                      |*             *
     *                | *           *
                      |   *       *
    - - - - - - - - - + - - - * - - - - - -
                      |     (2,0)
    *                 |
                      |
 
Re: drawing a curve based on derivatives of piecewise functi

SilentSymphony said:
im not asking for the answer, i just would like to know how to start working on this problem... any guidance yall can give would be awesome! :D
 
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