Jumpstart - Questions and Answers Week 4

Question 1 (Wes Whiting, incoming 2018)

I am familiar with the version of the Mean Value Theorem which the lectures attribute to Lagrange. However, I've never seen the more general version of it given afterward, and it's not obvious to me what makes it a useful generalization. Is there a natural geometric interpretation like for Lagrange's version?

I learned the more general version as Cauchy's Mean Value Theorem. Beyond recovering Lagrange's formula by taking $g(x) = x$, I'm not sure right now what Cauchy's version is used to solve (I'm sure there are theorems and problems in which it would be helpful, I just can't think of any at the moment). I think the geometric interpretation is similar to Lagrange's version. For Lagrange, MVT tells us that there is a point $c$ for which the tangent to [the graph of] $f$ at $c$ [$\bigl( c,f(c)\bigr)$] is parallel to the secant line through the endpoints of [the graph of] $f$ on whatever closed interval we have. For Cauchy, if we have $f$ and $g$ satisfying the hypotheses on an interval $[a,b]$, we can consider the curve parameterized by $x = f(t)$ and $y = g(t)$ for $t \in [a,b]$. Then there should be a point $c \in (a,b)$ for which the tangent at $C=\bigl(f(c),g(c)\bigr)$ is parallel to the line through the endpoints of the curve $A=\bigl(f(a),g(a)\bigr)$ and $B=\bigl(f(b),g(b)\bigr)$.

Question 2 (Michael Schirle, incoming 2018)

I'm worried I'm missing something in the second question because I've realized I'm not using all the assumptions in my attempted solution. Is there an obvious example of a differentiable function $f$ with $\lim_{x\to\infty}f(x)=0$ but $\lim_{x\to\infty}f'(x)\not=0$? Obviously, by the way the question is posed, such an $f$ would have an unbounded second derivative. I'm having trouble thinking of an example of such a function, and I think having such an example would give an insight into where my mistake is.

Imagine a sawtooth function where the size of the teeth shrinks toward zero, but the slope of each tooth is the same. Then round off the corners so it's differentiable everywhere. For a more concrete example, try something like $\frac{1}{x}\sin(x^2)$.