The equation of a curve is \(y = x(3 - x^{2})\). Find the equation of

Explanation

Curve : \(y = x(3 - x^{2})\)

\(y = 3x - x^{3}\)

Gradient function : \(\frac{\mathrm d y}{\mathrm d x} = 3 - 3x^{2}\)

When x = 2, y = \(3(2) - (2^{3}) = -2\)

Gradient of tangent : \(3 - 3(-2)^{2} = -9\)

Gradient of normal : \(\frac{1}{9}\)

Equation of normal through (2, -2)

\(\frac{y + 2}{x - 2} = \frac{1}{9}\)

\(9y + 18 = x - 2 \implies x - 9y - 20 = 0\)