CPM Homework Help : CALC Problem 3-153

Sketch a continuous curve which meets all the criteria:

Hint 1:

$f^\prime(x) > 0$ means $f(x)$ is always increasing.

Hint 2:

$f(x)$ looks something like a square root graph (w/out the endpoint) or a logarithmic graph (w/out the asymotote).

Hint 3:

If $f(2) = 1$ , then the point $(2,1)$ lies on the graph of the function.

How many roots do $f(x)$ have?
Hint (a):
Is it possible for a function that is always increasing (hint 1) AND always concave down (hint 2) to have no roots?
What can you say about the location of the root(s)?
Hint (b):
On what domain could the root(s) not exist?
Find $\lim\limits_ { x \rightarrow - \infty } f ( x )$ .
Hint (c):
Could there be a horizontal asymptote as $x → −∞$ ?
Is it possible that $f^\prime(1) = 1$ ? $f ^ { \prime } ( 1 ) = \frac { 1 } { 4 }$ ? For each case, explain why or why not.