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can you explain these questions in detail? By writing all the details on the paper for each question. Thanks ..:)

1. Assuming , we have

so that

2. Let . The gradient of is

and the gradient at the desired point is

The tangent plane then has equation

3. has critical points where the derivatives vanish or do not exist. The latter is irrelevant here, since is a polynomial and is thus continuous everywhere. So just differentiate and set the derivative equal to 0:

The last equation tells us , , or .

If , then in the first equation we get .

If , then .

If , then .

So there are 3 critical points at (0, 0), (1, -2), and (1, 2).

4. Stationary points are critical points where the derivative vanishes. So same process as in (3):

The first equation tells us or . for all , so we can ignore this term.

If , then .

If , then .

The second equation tells us or . But this gives the same information as before.

So there are only two stationary points, (0, 0) and (-1, -1).

5. First find the stationary points:

Solve this to find the one point at (0, 0). At this point, we have .

Now check for extrema along the boundary. The rectangle is bounded by the lines , , , and .

On , we have

which attains a minimum of 3 when and a maximum of 7 when .

On , we have

which has a minimum of 3 at and a maximum of 7 at

We can check on the other 2 lines, but we'd get the same information.

So we've found that has an absolute minimum of 0 at (0, 0), and an absolute maximum of 7 at both (-2, -1) and (2, 1).

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the answer is d on edge

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