Question a) b) Let f(x, y) x²+2y Find the first partial derivatives of f. Determine whether the limit lim f(x, y) exists or not. (x,y) →(2,-2) c) Determine whether the function [ f(x, y), (x, y) = (0,0) is g(x, y) = {1, (x,y) = (0,0) {F(x, −1, continuous at the points (2,-2) and (0,0).
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- There is a function f(x,y)=3√x3 + y. a)Calculate the partial derivative f,x (x, y) at points different from the point [0, 0] in which it is defined.Partial derivatives from the definition Suppose ƒ(x, y) = x2y. Use the limit definition of partial derivatives to compute ƒx(x, y) and ƒy(x, y).22) Find the first partial derivatives of the function. f(x, y) = x^y
- A function f has continuous second partial derivatives on an open region containing the critical point (a, b). If fxx(a, b) and fyy(a, b) have opposite signs, what is implied? Explain.Let f be a function that admits continuous second partial derivatives such that ∇f (x, y) = (ax2 - x, y2 - a2) with a <0. It can be stated with certainty that:A) The point (1/a, a f(1/a, a)) is a saddle point of f and f reaches a relative maximum at the point (0, a).B) The point (1/a, a, f(1/a, a)) is a saddle point of f and f reaches a relative maximum at the point(-1/a, a).C) The point (0, a, f(0, a)) is a saddle point of f and f reaches a relative minimum at the point (1/a, a).D) The point (0, −a, f(0, −a)) is a saddle point of f and f reaches a relative minimum at the point(0, a).Show that the function f:R^(2)->R defined by f(x,y)={((xy^(2))/(x^(2)+y^(4)), if x^(2)+y^(4)!=0),(0, if x=0=y):} possess first order partial derivatives everywhere including the origin but the function is discontinuous at the origin