Find the stationary points of the function z = 3x(x+y)3 - x3 + 24x
z = 3x(x+y)3 - x3 + 24xDifferentiating partially with respect to x and with respect to y:∂z/∂x = 3(x+y)3 + 9x(x+y)2 - 3x2 + 24∂z/∂y = 9x(x+y)2At stationary points: ∂z/∂x = 0 and ∂z/∂y = 0.From ∂z/∂y = 0 we deduce: x = 0 or y = -x.We consider ∂z/∂x = 0 in each of these cases:For x = 0:3y3 + 24 = 0y = -2Hence a stationary point at (0, -2, 0)For y = -x:-3x2 + 24 = 0x = 2√2 and x = -2√2Hence stationary points at (2√2, -2√2, 32√2) and (-2√2, 2√2, -32√2)
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