Question

Simultaneous equations and differentiation?

Variables `u`, `x` and `y` are such that `u = 2x(y-x)` and `x + 3y = 12`. Express `u` in terms of `x` and hence find the stationary value of `u`.

Answer 1

Please see below.

Explanation:

Solve `x+3y=12` for `y`

`y = 4-x/3`

Substitute in the expression for `u`

`u = 2x(4-x/3-x) = 2x(4-(4x)/3) = 8x-(8x^2)/3`

Differentiate

`(du)/dx = 8-(16x)/3`

Find `x` values where `(du)/dx = 0`

`x = 3/2`.

If stationary value" means the `u` value, use `u = 8x-(8x^2)/3` to find it.