Find two positive numbers whose sum is 100 and whose product is a maximum.
Call the two numbers x and y. The constraint is that x + y =100, and we need to maximise A=xy.
Rearrange the constraint to y = 100 - x, and substitute into the product equation.
A = x(100-x) = 100x - x2
Differentiate to find the critical points:
A' = 100 - 2x = 0100 = 2xx = 50
Differentiate again to check that this is indeed a maximum.
A'' = -2
The second derivative is always negative so A = 100x - x2 is always concave so the critical point is indeed a maximum.
Now it is easy to find y since we have x. y = 100 - 50 = 50
So the answer is x = 50 and y = 50.
Related Maths A Level answers
All answers ▸The equation of a circle is x^2-6x+y^2+4y=12. Complete the square to find the centre and radius of the circle.
(a) Express (1+4*sqrt(7))/(5+2*sqrt(7)) in the form a+b*sqrt(7), where a and b are integers. (b) Then solve the equation x*(9*sqrt(5)-2*sqrt(45))=sqrt(80).
Find the turning value of the following function, stating whether the value is min or max, y = x^2 -6x + 5
How many books and modules and what are they all about?