Question about linearization. Is this estimate is too large or small? 

I have done both parts, but I would just like to make sure. 
a.) Gives $f(1.9)\approx 4.7$ and $f(2.1)\approx 5.3$ (by $f(x)\approx L(a)=f(a)+f'(a)(x-a)$ we know this when $x$ is near $2$).
I'm not so sure about the second part. Here's what I think though, since the graph of $f'$ behaves by concaving downwards and decreasing, $f$ must be concaving upwards and increasing? Thus the tangent must be underneath the concaved curved, and our estimates fall short? Thoughts?
 A: Your approach to the problem is largely correct, but you've got the wrong picture. Try phrasing things more formally to make the big picture clearer, like so:

Since $f'$ is strictly decreasing on this interval around $x=2$, $f''$ must be negative, meaning $f$ is concave. Therefore the graph of $f$ around $x=2$ lies under the tangent line to the graph at $x=2$. (This is the definition of concave.) Therefore the linearization is an overestimate to the function at $x=2$.

The things to notice are:


*

*The fact that $f'$ is positive doesn't matter here. All we're looking for is whether $f$ is convex or concave; it doesn't matter whether or not $f$ is increasing. I think you may have confused yourself with the idea of "concaving upwards." You should draw examples of increasing and decreasing concave functions (and convex functions) to get a feel for what these definitions really mean.

*Since the only important information is whether or not $f$ is concave, you should just deduce what you can about the second derivative.
