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For a function $f(x)$ that is concave down (such as the logarithmic or square root functions), and an x-value of $c$, which if the following would be true statements? (There may be more than one correct answer.)

1) The magnitude of the error in any tangent line approximation would be proportional to the distance away from $c$. That is, the further away from $c$, the greater the error.

2) If the function is concave down, then the slope of the tangent line approximation must be negative.

3) Any tangent line approximation would over-estimate function values to the right of $c$ but under-estimate function values to the left of $c$.

4) Any tangent line approximation would always under-estimate function values.

5) Any tangent line approximation would always over-estimate function values.

I don't even know where to start which these! I do believe that number 2 is wrong, since if the function is concave down then the tangent line would have a positive slope, at least if it is like a square root function. Any suggestions as to what the others mean? Thank you!

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If the graph of a function is concave down, then slopes of tangent lines could be negative or positive (for the square root function, they are positive). Consider the graph of $y=\sin x$ over $[0,\pi]$, for example. So 2) is false. For the other items, try drawing a curve that is concave down (I suggest the graph of $\sin$ over $[0,\pi]$), draw a few tangent lines, and see what you can deduce. –  David Mitra Jan 20 '13 at 2:22
    
Note for a differentiable function whose graph is concave down, tangent lines always lie above the graph. So if you estimate a value $f(x)$ with the corresponding value of the tangent line at $x$, what can you say? This should help you with 3), 4), and 5). But also note the estimate would be exact for $x=c$... –  David Mitra Jan 20 '13 at 2:31

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