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 ▸How do I remember the trigonometry identities from C3 in the exam?
An object of mass 2kg is placed on a smooth plane which is inclined at an angle of 30 degrees from the ground. Calculate the acceleration of the object.
Why does 'x' need to be in radians to differentiate 'sin x'?
A ball is thrown from ground level at an angle of 30 degrees from the horizontal with a velocity of 20 m/s. It just clears a wall with a height of 5m, from this calculate the distances that the wall could be from the starting position.