### Home > CALC > Chapter Ch3 > Lesson 3.4.1 > Problem 3-153

3-153.

Sketch a continuous curve which meets all the criteria: Homework Help ✎

*f*′(*x*) > 0 for all*x**f*(*x*) is concave down.*f*(2) = 1

How many roots do

*f*(*x*) have?What can you say about the location of the root(s)?

Find

. Is it possible that

*f*′(1) = 1 ?? For each case, explain why or why not.

*f*'(*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.

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

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

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