Quiz 8 is here. Please remember that the second midterm is this Wednesday.

Solutions follow.

Problem 1 asks us to determine the local maxima and minma of .

We first note that the function is continuous and differentiable everywhere, except at , where is not even defined. We have

using the quotient rule. Since for all , it follows that the only solution to is . Since is and is , it follows that the only extreme point of occurs at , and it is a minimum.

This completes the problem, but it may be instructive to analyze the function a bit more.

Note that

and ,

so the -axis is a vertical asymptote at .

Also,

so the -axis is a horizontal asymptote as , while, using L’Hôpital’s rule, we have

.

Note that if and if . Also, from the sign of the derivative, is decreasing in and in , and it is increasing in .

Finally,

Clearly, if and if .

This means that is concave down in and concave up in .

Combining these observations allows us to sketch with reasonable accuracy. The graph of the function is shown below.

(Click the graph to enlarge.)

Problem 2 asks us to consider the function , and to indicate the intervals where it is increasing or decreasing, concave up, or concave down.

This function and its derivative are polynomials, so they are defined for all values of . First, we identify the values where .

We have

Next, we identify the values where .

We have

so , or , so or .

Note that , i.e., the zeroes of and of intertwine. (This is actually a general phenomenon.)

To determine the concavity of in the intervals determined by the zeroes of , we evaluate at points in the intervals and . Thanks to the second derivative test, we can save some time, and use these evaluations to (simultaneously) determine the extrema of and the intervals where increases and decreases.

We have , so is a local maximum of , and is concave down in .

Similarly, so is a local maximum of , and is concave down in .

Finally, , so is a local minimum of and is concave up in .

Moreover (due to the placement of its extreme points) we can also deduce that is increasing in decreasing in increasing in and decreasing in , so give us not just local maxima but global maxima of

The graph of is shown below.

(Click the image to enlarge).

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Sunday, October 24th, 2010 at 1:41 pm and is filed under 170: Calculus I. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

Craig: For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the current records are. Perhaps the first pape […]

Yes. Consider, for instance, Conway's base 13 function $c$, or any function that is everywhere discontinuous and has range $\mathbb R$ in every interval. Pick continuous bijections $f_n:\mathbb R\to(-1/n,1/n)$ for $n\in\mathbb N^+$. Pick a strictly decreasing sequence $(x_n)_{n\ge1}$ converging to $0$. Define $f$ by setting $f(x)=0$ if $x=0$ or $\pm x_n […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

It is easy to see without choice that if there is a surjection from $A$ onto $B$, then there is an injection from ${\mathcal P}(B)$ into ${\mathcal P}(A)$, and the result follows from Cantor's theorem that $B

Only noticed this question today. Although the selected answer is quite nice and arguably simpler than the argument below, none of the posted answers address what appeared to be the original intent of establishing the inequality using the Arithmetic Mean-Geometric Mean Inequality. For this, simply notice that $$ 1+3+\ldots+(2n-1)=n^2, $$ which can be easily […]

First of all, $f(z)+e^z\ne 0$ by the first inequality. It follows that $e^z/(f(z)+e^z)$ is entire, and bounded above. You should be able to conclude from that.

Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$. Of course there are many possible choices […]

I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is no […]