This is MULTIVARIATE Calculus question. PLEASE DO NOT PROVIDE text book extracts – NOT acceptable. No standard solutions acceptable either. Any solution has to address the question exactly.

If you do not have any MULTIVARIATE calculus background- DO not waste your time – with political science degree background. Thank you.- see attached file for further clarity.

1.Consider the function

f(x,y)=f(x,y) =x^3/3 −yx^2/2 −αx^2/2+ αyx +2y^3/3+ y^2

where α ≥ 0 is an unspecified coefficient. There are two values of α for which f(x,y) has three distinct critical points. The first of these is α = 0, where the critical points of f(x,y) are (0,0), (0,−1) and(-4/3,-4/3) . The second we will denote α∗.1

a) Determine the value of α∗ > 0 for which f will have three distinct critical points.

b) Use the second derivative test to characterize the critical points of f when α = 0.

2. Use the method of Lagrange multipliers to determine the minimum value of

f(x,y,z) =x^2+y^2+z^2 subject to the constraints

x + 2z = 6 and x + y = 12

State the minimum value of f and the associated values of x, y and z.

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