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)
`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:
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"
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