G.3.a.i)

When you go with , you can spot the maximizer at .

Look at this plot of at various points on the circle of radius centered at the maximizer at :

Why do you think that those scaled gradient vectors are pointing the way they are?

G.3.a.ii)

When you go with , you can spot the minimizer at .

Look at this plot of at various points on the circle of radius centered at the minimizer, :

Why do you think that those scaled negative gradient vectors are pointing the way they are?

G.3.b.i) The heat seeker

The Vector Calculus&Mathematica Missile Company is working on some primitive heat seeking devices and you are chief engineer of the TAD (Target Acquistion Division). The current problem under study is to program a device to go to  the hottest point in a temperature distribution.  
For instance, if
       
measures the temperature at a point , then the hottest point is because the denominator is smallest at this point.
You can use the gradient to try to make a heat seeking device that starts at and tries to seek out the hottest spot .
Here's a look:

The heat seeker can't tell the exact location of the hot spot, but it can sense the gradient of the merely by noting the hottest direction at a point .
If the heat seeker is at a point , why should you program the heat seeker so that it leaves in the direction of ?  

G.3.b.ii)

Given:
-> The heat seeker can update its direction every instant.
-> The heat seeker is programmed so that it leaves in the direction of .
Explain why the following plot displays a good approximation of the heat seeker's actual path when the seeker starts at :

G.3.b.iii)

Given:
-> The heat seeker can update its direction every instant.
-> The heat seeker is programmed so that it leaves in the direction of .
Give a plot of the heat seeker's path when the seeker starts at
.

G.3.c.i) Bad news

The people over at the assembly division tell you that the heat seeker can't be built so as to update its direction at every instant.  Instead, it will update its direction many times, but it will move on straight line segements between direction updates. Your group at TAD reacts to this information by programming the heat seeker as follows:
-> If the heat seeker is at , then the heat seeker moves to a new point
       
where the jump is positive number selected by trial and error.
For appropriately small jump numbers why is this a good update?

G.3.c.ii)

Now go from theory to practice.
Start at and program the heat seeker with and updates:

Good start, but the heat seeker lost its cool.  It nearly ran right over the hottest spot without even stopping to say hello.
Reprogram it with a smaller jump and more updates:

This time the heat seeker got close, but it blew its cool just as it was about to accomplish its desires.
Use trial and error to program in a jump size and an update number that send the heat seeker steadily to the hot spot so it can ignite its warhead and blow that hot spot to smithereens.

G.3.c.iii)

One way to increase the efficiency of the heat seeker is to use one of the larger jump sizes at first and run it until it goes bats.  Then use the last good point generated as a new starting point with a new, reduced jump size and run again.
Try this out starting at on the same function as above, incorporating any additional ideas that come to you.

G.3.d)

Use jumps and updates, as above, to guide the heat seeker starting at to the hottest point for:

You can see that the hottest point is
because and
make the denominator of as small as it can be.