### Home > CALC > Chapter 3 > Lesson 3.4.1 > Problem3-153

3-153.

Sketch a continuous curve which meets all the criteria:

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

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

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

• $f^\prime(x) > 0$ for all $x$

• $f(x)$ is concave down.

• $f(2) = 1$

• $f ^ { \prime } ( 2 ) = \frac { 1 } { 2 }$

1. How many roots do $f(x)$ have?

Is it possible for a function that is always increasing (hint 1) AND always concave down (hint 2) to have no roots?

2. What can you say about the location of the root(s)?

On what domain could the root(s) not exist?

3. Find $\lim\limits_ { x \rightarrow - \infty } f ( x )$.

Could there be a horizontal asymptote as $x → −∞$?

4. Is it possible that $f^\prime(1) = 1$$f ^ { \prime } ( 1 ) = \frac { 1 } { 4 }$? For each case, explain why or why not.