APPM 2350 Midterm: appm2350fall2014exam2_sol
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1. (a) not always true if a function has a local maximum at (a, b) then either. F|(a,b) = 0 or at least one of fx and fy does not exist. Consider the cone described by f(x, y) =px2 + y2. The function has a local minimum at (0, 0) but. F|(0,0) does not exist. (b) not always true consider the plane f(x, y) = x + y. Y for all (x, y) but f(x, y) is not a constant function. (c) not always true consider the following function f(x, y) =( xy. 0 x2 + y2 for (x, y) 6= (0, 0) for (x, y) = (0, 0) The partial derivatives of f clearly exist at (0, 0) because f is identically zero along the x and y-axes. But the function is not continuous at (0, 0). To see this note that along any line of the form y = mx we have f(x, y) m.
