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easy differentiation question

this is a easy question, but iam tired and can't think right at the moment.

how would u do:

1) two numbers have a sum of 20. use calculus to find the max product of the 2 numbers.

2) two numbers have a sum of 5. what is the min possible value of the sum of their squares

Reply 1

RichE

1) two numbers have a sum of 20. use calculus to find the max product of the 2 numbers.

Call them x and 20-x. Then their product is

S(x) = x(20-x) = 20x - x^2

dS/dx = 20 - 2x = 0 when x = 10.

So the maximum product is 100 when the two numbers are 10.

2) two numbers have a sum of 5. what is the min possible value of the sum of their squares

Call them x and 5-x. Then the sum of their squares is

T(x) = x^2 + (5-x)^2 = 25 - 10x + 2x^2

dT/dx = 4x - 10 = 0 when x = 2.5

So the minimum of the sum of their squares is 13.5 when both numbers are 2.5

Reply 2

ronnie

cheers !

u know how to do:

cost of providing a meal service on an airplane is $20 per passenger. the airline wants to set a price of $x per meal so that it makes a max profit. market research tells the airline that the number of passsenger on a flight prrepared to pay $x for a meal is given by the formula 45-x/2x
what price should the air line charge per meal?

AND

the cost of fuel per hour of running a jet aircraft, $C, is a function of its crusing velocity, v km/h:

c(v)= v^2+500 000/300

find the most economical crusing velocity for a journey of 6000km