5. (a) What are the conditions for a function of two variables f(x, y) to have...

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5. (a) What are the conditions for a function of two variables f(x, y) to have a critical point? Give sufficient conditions in terms of the second derivatives at the critical points for classifying them as local maxima, local minima and saddle points. (b) Find and classify all the critical point(s) of f(x, y) = xy x + 2x - (c) Write down a general expression for the Taylor expansion of an arbitrary func- tion of two variables, u(x, y), about a point (xo, yo), up to and including second (quadratic) order. Hence expand the above function f(x, y) about (0, 1) up to second order. (d) Let g(x, y) = f(x, y). Using the previous result write down the Taylor expan- sion of g(x, y) about (0, 1) to fourth order. (e) Show that g(x, y) has a critical point at (0, 1). Explain why the criteria of part (i) for classifying critical points does not apply here. Nevertheless, determine whether it is a maximum, a minimum or a saddle point, justifying your answer. 5. (a) What are the conditions for a function of two variables f(x, y) to have a critical point? Give sufficient conditions in terms of the second derivatives at the critical points for classifying them as local maxima, local minima and saddle points. (b) Find and classify all the critical point(s) of f(x, y) = xy x + 2x - (c) Write down a general expression for the Taylor expansion of an arbitrary func- tion of two variables, u(x, y), about a point (xo, yo), up to and including second (quadratic) order. Hence expand the above function f(x, y) about (0, 1) up to second order. (d) Let g(x, y) = f(x, y). Using the previous result write down the Taylor expan- sion of g(x, y) about (0, 1) to fourth order. (e) Show that g(x, y) has a critical point at (0, 1). Explain why the criteria of part (i) for classifying critical points does not apply here. Nevertheless, determine whether it is a maximum, a minimum or a saddle point, justifying your answer.