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  1. 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

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

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

    3. Find .

    4. 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 → −∞?