3. (a) Suppose :-/(z,y) describes the surface of a mountain. The valus of some d

Question

3. (a) Suppose :-/(z,y) describes the surface of a mountain. The valus of some derivatives are known at certain points: A 0 1 4 -3 2 C 10 2 5 3 2 4 3 For each situation, identify a suitable point and briefly explain: e.g. "X, because i. Mika wants to hike to point with view in every direction (where the mountain does Answer: it is a local minimum." Justify your answers with calculations. not obscure her view). Where shoulds he go? ii. Xiaofei is a biologist who wants to re-introduce a species of frog into a wet envi- ronment. Where is water mostly likely to form a pool? Answer: ili. Donnie wants to have picnic at a place where the ground is fat and dry. He thinks a peak would be too windy. Where should he go? Answer: iv. Akshat also wants to have a picnic where the ground is flat, but he likes surprises. Where should he go? Answer:

Explanation / Answer

We need to calulate D for each point where fx = fy = 0

D = fxx(a,b) fyy(a,b) - fxy2(a,b)

For B,

D = 4 * (3) - (-4)2 = -4

For E,

D = (-3) * (-3) - (2)2 = 5

For F,

D = 5 * (5) - (5)2 = 0

For G,

D = 3 * (4) - (3)2 = 3

Now,

For E, D > 0 and fxx(a,b) < 0, then f has a relative maximum at E.

For G, D > 0 and fxx(a,b) > 0, then f has a relative minimum at G.

For B, D < 0, so B is a saddle point.

For F, D = 0, so no conclusions can be drawn.

i)

Mike wants to go at the highest position. So Mike Should go to E.

ii)

Water will form pool at the lowest position. So Water will form pool at G.

iii)

Donnie should go to point which is a saddle point. Donnie should go to B.

iv)

Akshat should go to a point where we do not know whether it is a local minima or maxima. Akshat should go to F as it is local maxima/minima but we do not know whether it is maxima or minima.

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