Sketch a graph with the given parameters

[Ric]

What I understand is:
f(-infinity,1) Slope is positive
f(1,3) Slope is negative
f(3,infinity) Slope is positive
f '(1)=undefined
f ''(1)=undefined
f '(3)=0

What I do not understand is the last line.

Also if the slope is positive on (3,infinity), how can the limit be -infinity as x approaches infinity?

 

[soroban]

Hello, Ric!

I agree with your assessment.
There are obviously typos in the problem.
I conjecture the following corrections on conditions (5) and (6).

Sketch the graph of a ?continuous? function that satisfies all the given conditions.
Describe how the conditions determine the graph.

\(\displaystyle (1)\;f'(x) < 0 \text{ if }1 < x < 3\)
\(\displaystyle (2)\;f'(x) > 0 \text{ if }x < 1\text{ or } x > 3\)
\(\displaystyle (3)\;f'\text{ and }f''\text{ are unde{f}ined at }x = 1.\)
\(\displaystyle (4)\;f'(3) \,=\,0\)
\(\displaystyle (5)\;f'' > 0\text{ for }x \ne 1\)
\(\displaystyle \displaystyle(6)\,\lim_{x\to\infty}f(x) = \infty\)

Condition (3) cannot be fulfilled if the graph is continuous.

Conditions (1) and (2) tell of the slope on those intervals.
Condition (3) suggests there is a vertical asymptote at \(\displaystyle x = 1.\)

The graph might look like this:

. . \(\displaystyle \begin{array}{ccccc} & : \\ \nearrow & : & \searrow && \nearrow \\ -- & * & -- & * & -- \\ & 1 && 3 \end{array}\)


(4) says there is a horizontal tangent at \(\displaystyle x = 3.\)
. . There is a minimum at \(\displaystyle (x,\,f(3))\)

(5) says the graph is always concave up (except at \(\displaystyle x = 1\))


I would guess that the graph might look like this:

          |
          |   :
          |  *:*                *
          |   :
          |   :
          | * : *              *
          |   :
          |*  :  *            *
          *   :   *          *
        * |   :     *      *
    *     |   :         *
          |   :
  - - - - + - + - - - - + - - - - - -
          0   1         3

 

[Ric]

Thank you both for all your help and insight.

this was the correct parameter



and also lim -> infin f(x) = infin.