Question

A continuous function f, defined for all x, has the following properties:

1. f is increasing
2. f is concave down
3. f(13)=3
4. f'(13)=1/4

Sketch a possible graph for f, and use it to answer the following questions about f.

A. For each of the following intervals, what is the minimum and maximum number of zeros f could have in the interval? (Note that if there must be exactly N zeros in an interval, the minimum and maximum are both N.)


−INF <x<= 0
0 <x<=1
1<x<13
13<=x<INF
I need the maximum and minimum... I just have the two first ones... 0 for both 1 and 2 maximum and minimum... then I really don't know what to do..

B. Are any of the following possible values for f'(1)? (Enter your answer as a comma-separated list, or enter 'none' if none of them are possible.) −3, −2, −1, −51, 0, 51, 1, 2, 3.
possible values: f'(1)=_________

C. What happens to f as x−>- INF?
lim x−> INF f(x)= ________

(Enter the value, 'infinity' or '-infinity' for or −, or 'none' if there is no limit.)

I realy don't know how to do these problems.. please help

Answer 1

For a function to be increasing on ℝ and concave downwards, it must:
1. have no maximum/minimum, because it is a one-to-one function.
2. Crosses the x-axis (or any other y-value) <i>at most</i> once, since it is an increasing function.
3. Since it is concave down, d²y/dx² must be negative.

Study the above statements. If you do not know why they are true, refer to your notes, your textbook, or post.

I have taken an example function that satisfies these two properties is f(x)=3-e^(-x). (There may be many other functions that have these properties.) See the following link for its graph:
http://img268.imageshack.us/i/1298419029.png/

A. Since we know that f(13)>0, then the zero of the function, <i>if any</i>, must happen when x<13 for an increasing function. Beyond 13, x continues to increase, and therefore cannot have zeroes.

B. If f'(13)=1/4, and we know that f"(x) is negative throughout the domain of f(x), what can you say about f'(x) for x>13? Would it not be true that f'(x)<1/4 for x>13?
So what happens to to f'(x) <13?

C. The limit for the example function is 3, (but f(x) does not satisfy the required conditions of f(13)=3 and f'(13)=1/4.) Some parameters are required. It was just for illustration purposes.