drawing a curve based on derivatives of piecewise function

[SilentSymphony]

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!

 

[pka]

Re: drawing a curve based on derivatives of piecewise functi

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.

 

[soroban]

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 \(\displaystyle y\:=\:f(x)\) with:

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

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

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

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

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

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

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

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

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

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

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

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


The graph must look something like this:

           (-2,8)     |
              *       |
          *       *   |
        *           * |                *
       *             *|
                      |
      *               *(0,4)          *
                      |
                      |*             *
     *                | *           *
                      |   *       *
    - - - - - - - - - + - - - * - - - - - -
                      |     (2,0)
    *                 |
                      |

 

[pka]

Re: drawing a curve based on derivatives of piecewise functi

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!

 

[SilentSymphony]

thanks. ya'll are awesome! :lol: