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.
Cheers
As a typical example, say you have 30 m of fencing to surround a rectangular area and you need to find the maximum such area. You have the equation
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 $\displaystyle x^3 + y^2$. It's the same process.
-Dan
OK, you are wanting to minimise $\displaystyle \displaystyle \begin{align*} f(x, y) = x^3 + y^2 \end{align*}$ subject to $\displaystyle \displaystyle \begin{align*} g(x, y) = x + y = 4 \end{align*}$. To minimise this we require using Lagrange Multipliers, i.e. to solve the equations
$\displaystyle \displaystyle \begin{align*} \nabla f(x, y) &= \lambda \nabla g(x, y) \\ \left( 3x^2 , 2y \right) &= \lambda \left( 1, 1 \right) \\ \\ \begin{cases} 3x^2 &= \lambda \\ 2y &= \lambda \end{cases} \\ \\ 2y &= 3x^2 \\ y &= \frac{3}{2}x^2 \end{align*}$
And since we know by our constraint that $\displaystyle \displaystyle \begin{align*} x + y = 4 \end{align*}$, that means
$\displaystyle \displaystyle \begin{align*} x + \frac{3}{2}x^2 &= 4 \\ 3x^2 + 2x &= 8 \\ 3x^2 + 2x - 8 &= 0 \\ 3x^2 + 6x - 4x - 8 &= 0 \\ 3x \left( x + 2 \right) - 4 \left( x + 2 \right) &= 0 \\ \left( x + 2 \right) \left( 3x - 4 \right) &= 0 \\ x = -2 \textrm{ or } x = \frac{4}{3} \end{align*}$
Since x and y have to be positive, that means $\displaystyle \displaystyle \begin{align*} x = \frac{4}{3} \end{align*}$ and $\displaystyle \displaystyle \begin{align*} y = \frac{8}{3} \end{align*}$.
You can find the minimum value by subbing into the original function.