Expert Help with High School Geometry

I'm having trouble understanding how to check the second order conditions for my unconstrained maximization problem.

This is the entire problem: Alicia wants to maximize her grade, which is a function of the time spent studying ($T$) and the number of cups of coffee ($C$) she drinks. Her grade out of 100 is given by the following function.
$G(T,C)=50+10T+16C-<!--\; -\; -->(T^2+2TC+2C^2)$
In the first order conditions, I find the partial derivatives and set them equal to zero. I get the following two equations:

$10-<!--\; -\; -->2T-<!--\; -\; -->2C=0$ and $16-<!--\; -\; -->2T-<!--\; -\; -->4c=0$. The first equation was the partial derivative with respect to $T$ and the second equation was the partial derivative with respect to $C$. Solving these two equations, I find that $C=3$ and $T=2$.

Now, I need to check the second order conditions. I know that the second partial derivative with respect to both $T$ and $C$ should be negative. This checks out. I get -2 from the first equation (with respect to $T$) and I get -4 from the second equation (with respect to $C$). The last thing I need to do with the second order condition is multiply these two together (which yields 8) and then subtract the following:
${\left(\frac{{\mathrm{\delta <!--\; \delta \; -->}}^{2}G}{\mathrm{\delta <!--\; \delta \; -->}T\mathrm{\delta <!--\; \delta \; -->}C}\right)}^2$
Please forgive me if this formula isn't displaying correctly. I tried using the laTex equation editor, but I'm not sure if it worked. Anyway, I need to know how to derive this. What is it asking for? I know that this part should be -2 squared, which is 4. Then, 8-4=4, which is positive and tells me that the second order conditions are met.

But where is the -2 coming from? I know within both of the equations, there are a few -2's. But, I'm not sure exactly where this -2 comes from.

Let the endpoint of a differentiable, vector-valued function a$(t):(t_a..t_b)\to <!--\; \to \; -->\mathbb{R}$ move on a circle in an Euclidean plane. Let $t_0\in <!--\; \in \; -->(t_a..t_b)$. Let $s(t)$ be the length of the arc between $t_0$ and point a$(t)$. Let there be a positive real number $\mathrm{\delta <!--\; \delta \; -->}$ such that $\mathrm{\forall <!--\; \forall \; -->}t:0<|t-<!--\; -\; -->t_0|<\mathrm{\delta <!--\; \delta \; -->}:s(t)-<!--\; -\; -->s(t_0)\ne <!--\; \ne \; -->0$.
My (physics) book uses a proof which seems to imply that $s$($t$), where $t$ is time, is differentiable in respect to $t$. How to prove its differentiability?

Just come across a question regarding sequential maximization and simultaneous maximization, and I do not recall whether there are any established conditions for the equivalence. Anyone has some idea?
$\underset{x}{max}\underset{y}{max}f(x,y)=\underset{y}{max}\underset{x}{max}f(x,y)?$

My teacher said that the central angles of a circle are equal to the measure of the arc, but I don't understand on how this could possibly work.
Can someone please explain how this is possible?

My initial investment is $\mathrm{\$<!--\; \$\; -->}100$, and I earn $1\mathrm{\%<!--\; \%\; -->}$ interest per day. I can opt for any number of compoundings per day (if twice per day, then the interest rate per compounding period is $0.5\mathrm{\%<!--\; \%\; -->},$ and so on), but I have to pay $\mathrm{\$<!--\; \$\; -->}0.01$ each time my interest is compounded. After 365 days I will close the account.
What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

Are partial derivatives of parametric surfaces always orthogonal?
For a surface $\mathbf{r}=\mathbf{r}(s,t)$ are the partial derivates ${\mathbf{r}}_s$ and ${\mathbf{r}}_t$ in general orthogonal?

my intuition says I'm right, but I couldn't find or get a proof for it:

Suppose I have a vector function $\stackrel{\to <!--\; \to \; -->}{x}(t)\in <!--\; \in \; -->{\mathbb{R}}^n$. I am interested in finding ${\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}_t||\stackrel{\to <!--\; \to \; -->}{x}(t)|{|}_2$. For some reasons, this optimization problem is hard to solve, what I can solve easily, however, is ${\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}_t||\stackrel{\to <!--\; \to \; -->}{x}(t)|{|}_1$.

I know that 1-norm and 2-norm are equivalent in ${\mathbb{R}}^n$. Does this also include that the solution of these two problems are equivalent?

So, I have an arc that's part of a circle that I must "travel across".

The circle has a radius of 15 - a circumference of 94.248 (rounding). The arc length in question is equal to 15.708.

Now, normally, you could sort of walk along the arc, turning with it. But, if I could only travel straight, I need to know how I would get from one part of the arc to 15.708 units later in the arc of the circle (only being able to travel straight, or make 90 degree turns).

I am writing a computer program where I have $x$ real positive varying in the domain $[\sqrt{U},U]$. I want the value of $x$ which maximizes:
$(1+\sqrt{U})-<!--\; -\; -->\frac{\sqrt{U}-<!--\; -\; -->1}{U-<!--\; -\; -->\sqrt{U}}x-<!--\; -\; -->\frac{U}{x}$
("visually", it seems to be a bit above $2\sqrt{U}$) what is the best way to find it exactly?

What is the height of a circular cone with surface line ("Mantellinie") s, which has the maximal volumina? My problem here is, that I am not really sure what surface line is..is this the same thing as just to say surface?

Pythagorean Theorem intuition
I actually have a degree in Pure Mathematics, and this has always bugged me. Like a tiny, hardly-noticeable stone in your shoe you just can't get out.
Despite being at the core of geometry, arithmetic, and infinity, despite being able to prove it a dozen different ways, and despite it being the first "formula" anyone learns and one of the oldest known mathematical ideas - I can only see shadows and silhouettes of its truth.
Looked at from the point of view of geometry, I can see the statement plainly as a statement regarding the intrinsic notions of angle and area (using the proof of similar right-angle triangles) - however, the last step requires a correspondence between the notions of length and area to make complete (namely, the area of a shape increases quadratically as its scale increases linearly). This has always bothered me, since, the Theorem is an idea solely between angle and length.
There are, of course, simple proofs that don't depend on the notion of area (ex: Pythagorean Theorem Proof Without Words 6), however, sadly, they don't further my intuition, despite being clever/cute.
Staring at the statement blank in the face, I understand it far less intuitively then I do half of the ideas in algebra, analysis, number theory, etc.

How to prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere?
The key to solving the problems is that we must take all curves connecting the two points , not just great-circle or parametrized differentiable curve, into account!
A similar problem is the straight line distance(shortest distance between two points): we must take all curves connecting the two points into account!

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document I'm having trouble finding out the first step to this problem.
So far all I have is:

Find parametric equations for the arc of a circle of radius 6 from $P=(0,0)$ to $Q=(12,0)$.

If segment mn is a diameter of a circle f, and v is a fixed point on mn, for which point E on MN is the length FE minimized?

Suppose that $f(\cdot <!--\; \cdot \; -->)$ is a quadratic and concave function such that it has a maximum, $x\in <!--\; \in \; -->\mathbb{R}$, $y\in <!--\; \in \; -->(0,+\mathrm{\infty <!--\; \infty \; -->})$. I have to solve a maximization problem that is
$\underset{{\displaystyle \frac{x}{y}}}{max}f(x/y)$
My question is that, since I want to maximize with respect to $x/y$, I will treat this as if $\frac{\mathrm{\partial <!--\; \partial \; -->}f(x/y)}{\mathrm{\partial <!--\; \partial \; -->}(x/y)}$?

Or do i need to maximize with respect to $x$, then in $y$ and use the Hessian matrix?

Or is it something else? Well it is a composition of functions after all, isn't it? I am a llitle confused...

Pythagorean Theorem for imaginary numbers
If we let one leg be real-valued and the other leg equal bi then the Pythagorean Theorem changes to $a^2-<!--\; -\; -->b^2=c^2$ which results in some kooky numbers.
For what reason does this not make sense? Does the Theorem only work on real numbers? Why not imaginary?

An ant starts on (0,3) on a circle with 3 radius and walks 2 units counterclockwise along the arc of a circle. Find the $x$ and $y$ coordinates of where the ant ends up.
I only know how to move in a straight line (up-down, left-right) to determine coordinates.
How would I begin to attempt this problem? I imagine I have to plug in a 2 into some sort of formula.

Verify the circumference of the circle formula for a circle of radius r using the arc length formula.
How do I complete this problem? Do I use the ellipse equation or the proper circle equation with (x, h) and (y,k)?

Approximation using pythagorean theorem
Im looking at the following diagram,

Here, I am interested in a relation between a,s,c, assuming that $r>>1$ my first thought is to relate then using the pythagorean theorem,
$c^2=a^2+s^2$
Is making sure an approximation a good approach and is there an alternate way to relate my variables? Thank you in advance!

$A(1,4),$ $B(2,3)$ are the vertices of $\mathrm{△<!--\; △\; -->}ABC$ .
Centroid of the triangle lies on the locus
$3x-<!--\; -\; -->9y+13=0$
If the equation of the locus of the third vertex C is $x-<!--\; -\; -->by-<!--\; -\; -->c=0$ then find $(b+c)$