Partial differentiation

In this question you must show detailed reasoning.

The surface S is given by the equation z = 4x²y - 3xy² - 5x for all real values of x and y.

The point P (1, 1, - 4) lies on S.

Find the equation of the tangent plane to S at P.

The point Q is a stationary point of S with positive x - and y-coordinates.

Find, in exact form, the coordinates of Q.

Find the value of the determinant of H, the Hessian matrix of S, at Q.

Deduce the nature of the stationary point Q.

The surface S has equation z = x sin y + for x > 0 and 0 < y < π.

Determine, as a function of x and y, the determinant of H, the Hessian matrix of S.

Given that S has just one stationary point, P, use the answer to part (a) to deduce the nature

of P.

The coordinates of P are (α, β, γ).

Show that β satisfies the equation β + tan β = 0.

Find the equation of the tangent plane to E at P, giving your answer in the form

ax + by + cz = d where a, b, c and d are integers.

For all real values of x and y the surface S has equation z = 4x² + 4xy + y² + 6x + 3y + k, where k is a constant and an integer.

Determine the smallest value of the integer k for which the whole of S lies above the x - y

plane.

The surface S has equation x² + y² + z² = xyz - 1.

Deduce that S has no stationary point.