CPM Homework Help : APCALC Problem 3-153

Sketch a continuous curve which meets all of the following 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 asymptote).

Hint 3:

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

Hint 4:

$f ^\prime(2) =\frac { 1 } { 2 }$ means the slope of the tangent line is at $x = 2$ is $\frac{1}{2}$

How many roots does $f$ 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?
What is $\lim\limits _ { x \rightarrow - \infty } f ( x )$ ?
Hint (c):
Could there be a horizontal asymptote as $x \rightarrow −\infty$ ?
Is it possible that $f^\prime (1) = 1$ ? $f ^\prime (1) =\frac { 1 } { 4 }$ ? For each case, explain why or why not.
Hint (d):
Remember: $f^\prime(2)=\frac{1}{2}$ and $f(x)$ is increasing and concave down.