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# Please do question 5a and 5b 4. In this problem we analyze the behavior of the...

Please do question 5a and 5b

4. In this problem we analyze the behavior of the polynomial f (x, y) = ax² + bxy + cy? (without using the Second Derivatives Test) by identifying the graph as a paraboloid. (a) By completing the square, show that if a + 0, then b 2 4ac - 62 f(x, y) = ax² + bxy + cy? = a [( 2 + Y + 2a 4a2 (b) Let D = 4ac – 62. Show that if D > 0 and a > 0, then f has a local minimum at (0,0). (C) Show that if D > 0 and a < 0, then f has a local maximum at (0,0). (d) Show that if D< 0, then (0,0) is a saddle point. 5. (a) Suppose f is any function with continuous second-order partial derivatives such that f(0,0) = 0 and (0,0) is a critical point of f. Write an expression for the second-degree Taylor polynomial, Q, of f at (0,0). (b) What can you conclude about Q from Problem 4?

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