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 ▸If f(x)=(4x^2)-(8x)+3, find the gradient of y=f(x) at the point (0.5,0)
Find the coordinates of the maximum stationary point of the y = x^2 +4x curve.
The second and fourth term of a geometric series is 100 and 225 respectively. Find the common ratio and first term of the series. Round your answer to 2 d.p if necessary
Curve D has equation 3x^2+2xy-2y^2+4=0 Find the equation of the tangent at point (2,4) and give your answer in the form ax+by+c=0, were a,b and c are integers.