Sketch the graph of a function that satisfies all of the conditions listed below.a. f(-x) = -f(x) b. f(0) = 0 c. lim x-->2  f(x) = - infinity d. lim x-->infinity  f(x) = 0 e. f''(x) < 0...

Sketch the graph of a function that satisfies all of the conditions listed below.

a. f(-x) = -f(x)

b. f(0) = 0

c. lim x-->2  f(x) = - infinity

d. lim x-->infinity  f(x) = 0

e. f''(x) < 0 on the intervals (0,2) and (2, infinity)

Asked on by bdremm

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted on

 

`f(-x) = -f(x)` means the function is "odd"

Visually, it means that the left side of the graph is a mirror image of the right side, but then flipped upside down.  For example:

 

Next `f(0)=0`

This already has to be the case, in order to satisfy the first condition (odd function), so we don't have any new info from this

 

Next: `lim_(x->2) f(x) = -oo`  

 

This also tells us (because of the odd function condition) that

`lim_(x-> -2) f(x) = oo`

 

Our graph has vertical asymptotes:

 

We can actually connect these up to complete the middle of our graph:

 

This graph also happens to be concave down on (0,2)

That is, `f''(x)<0`

on the interval (0,2), so we have satisfied that condition as well

By the way, the function I used to get that graph was: `x/(x^2-4)`  

 

Finally, as `x->oo` we want the graph to get close to 0, but we also want the graph to "curve down"

So:

 

Our odd function condition means that we must have:

The function I used here was  

`(10)/(x^2)` on the left side and `(-10)/(x^2)` on the right side

 

 

 

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