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

3-153.

Sketch a continuous curve which meets all of the following criteria:

• $f ^\prime (x) > 0$ for all $x$

• $f$ is concave down.

• $f(2) = 1$

• $f ^\prime(2) =\frac { 1 } { 2 }$

$f^\prime(x) > 0$ 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 asymptote).

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

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

1. How many roots does $f$ 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. What is $\lim\limits _ { x \rightarrow - \infty } f ( x )$?

Could there be a horizontal asymptote as $x \rightarrow −\infty$?

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

Remember: $f^\prime(2)=\frac{1}{2}$ and $f(x)$ is increasing and concave down.