t point) A bird in Its natural habitat feeds on two kinds of seeds, whose nutrit

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t point) A bird in Its natural habitat feeds on two kinds of seeds, whose nutritional values are 5 calories per seed of type 1 and 3 calories per seed of type 2. Both kinds of seeds are hidden among Itter on the forest floor and have to be found. If the bird splits attention Into a traction for food type 1 and torfood type 2, then ts probablity of frdng a s ofthe given type is Assume that the bid paystaat ention to searching for seeds so that xi +-1 where 0Sxi S landos 1. The expected nutritional value of any seed found on the forest floor can be caloulated by multiplying the nutritional value of a seed type by the probablity of finding that seed type and adding these products up over all seed types a)Werite down an expression for the expected nutritional value V(x) gained by the bird when it spits its attention. Use the constraint on x,J to elilminate x and rename the variable xi as. (b) Find the A Local minimum. B. Local maximum. C.Nelther Absolute minima- Absolute maxima- Use your results to determine the optimal strategy of the bird to maximize the nutritional value of the seeds it can find. A. The bird should spend its attention on a combination of both types of food to get the most B. The bird should spend all its attention on food of type 1 and none on food of type 2 to get the most nutritional value. C. The bird should spend all its attention on food of type 2 and none on food af type 1 to get the most nutritional vatue. value. MacBook Air

Explanation / Answer

Here x1 + x2 =1 where 0 < x1,x2 < 1

Expected Nutrition value E(NV) P = 5 P1(x1) + 3 P2(x2)

5 * x13 + 3 * x25 = 5x13 + 3x25

(b) V(x) = 5x13 + 3x25

V(x) =5x13 + 3 (1-x1)5 = 5x3 + 3(1-x)5

dV(x)/ dx = 15 x2  + 15 (1-x)4

x2= (1 - x)4

x= +- (1 - x)2

+-x = (1-x)2

+-x = 1 + x2 -2x

x2-3x +1 = 0; x2 -x +1 =0

x = 1 - 0.618 = 0.382

V(x)critical = 5 * (0.382)3 + 3 * (0.612)5= 0.5363

This point is critical minima. as d(V)/dx is positve

(c) Absolute minima = 0.5363

Absolute minima would be for x1 =1 and x2 = 0

Absolute Maxima = 5

Here the bird shoudl concentrate all its attention on food of type 1 and none on food of type 2 to get the most nutrititonal value. Option B is correct.

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