Math 192, Prelim2April 14, 20051) a) (7 points) Give an example of a functionf(x, y) that has a local maximum at (0,0).Show that your example is valid.b) (7 points) Give an example of a functionf(x, y) that has a saddle point at (0,0).Showthat your example is valid.2) (11 points) Determine all the maxima and minima of the functionf(x, y) =x2+xyonthe region of points (x, y) satisfyingx2+xy+y2≤1.3) (11 points) Find the average distance from a point in the interior (or on the boundary)of a circle of radiusRto the center of the circle.4) (11 points) LetCbe a circle of radius 2 +√22centered at the origin.Find the area ofthe regionoutsideCbutinsidethe region bound byr= 2 + cosθ.5) The questions below are true/false. You need not give reasons.You will be penalized for wrong answers, just like the SAT’s. So do not guess!!a) (±3 points) Iff(x, y) is given and±uis a unit vector in two dimensions then the directionalderivative offin the direction of±uat (x0, y0) isfx(x0, y0)±i+fy(x0, y0)±j.b) (±3 points) Iff(x, y, z) is a scalar function then the expression grad(div(
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