Little stuck on this question. Any help greatly appreciated.
Find two positive numbers whose sum is 4 and such that the sum of the cube of the first is and the square of the second is as small as possible.
P = 2x + 2y = 30
and you need to maximize the area: A = xy. So you solve the perimeter equation for y: y = 15 - x, then put that into the area formula: A = xy = x(15 - x). Then you take the derivative and set it to zero, etc.
So you've got x + y = 4 and you want to minimize . It's the same process.
OK, you are wanting to minimise subject to . To minimise this we require using Lagrange Multipliers, i.e. to solve the equations
And since we know by our constraint that , that means
Since x and y have to be positive, that means and .
You can find the minimum value by subbing into the original function.