Given:

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f is a multivariable function defined by f(x, y) = x^3y – x^2y^2 where x and y are real variables

A.  Pick a specific point P0 = (a, b, f(a, b)) on the surface z = f(x, y), other than (0, 0, 0) or (3, 21, -3402).

Note: You may use whimsical values for a and b, such as the month and day of your birthday. For example, March 21st becomes a=3 and b=21, so the 3rd coordinate is f(3, 21) = -3402.

1.  Calculate the directional derivative of the vector in the direction of greatest increase of the surface at P0. Use algebra and/or calculus techniques and justify all work.

2.  Find a direction vector in which the directional derivative of f(x, y) at P0 is equal to zero. Use algebra and/or calculus techniques and justify all work.

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