SOLUTION: Please answer this optimization problems 1:find the number/s in the interval I=[1/2,3] such that the sum of the number and it's reciprocal is a minimum. 2:A wire measuring 12 ft

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Question 1135313: Please answer this optimization problems
1:find the number/s in the interval I=[1/2,3] such that the sum of the number and it's reciprocal is a minimum.
2:A wire measuring 12 ft long is cut into pieces. One piece is to be bent into a square and the other into a circle. How should the wire be cut so that the combined area is as large as possible?
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  They ask you to find the minimum of the function

        f(x) = x + 1%2Fx

    in the interval  I = [1/2, 3].



    Take the derivative, equate it to zero and find the root:


    f'(x) = 1 - 1%2Fx%5E2;   f'(x) = 0  is the equation  1 - 1%2Fx%5E2 = 0;

    as an equation, it is equivalent to  x%5E2 = 1,  which has only one root  x= 1  in the given interval.



    ANSWER.  x = 1.



To better illustrate the situation for you, I placed the plot below:



    graph%28+330%2C+330%2C+-1.5%2C+3.5%2C+-5.5%2C+5.5%2C%0D%0A++++++++++x+%2B+1%2Fx%0D%0A%29


        Plot y = x + 1%2Fx.