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- 2) Determine as 1st order partial derivatives of the functions z=2e 2 and sin(x) at the pointThere 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.Find the first partial derivatives with respect to x and y of f(x,y) = (4x-y)/(4x+y) at the point (x,y) = (2,3)
- 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).40) Find the indicated partial derivatives. f(x, y) = In(x + arctan(y/x); f_x(2, 3)Assume that all the given functions have continuous second-order partial derivatives. If z = f(x, y), where x = 9r cos(?) and y = 9r sin(?), find the following. pls add the exact answer at the end of the explanation
- Find all the second-order partial derivatives of the functions g(x, y) = cos x2 - sin 3y1) Suppose that f(x, y, z) = x^2 + y^2 + z^2 + 3xyz. The equation x^2 + y^2 + z^2 + 3xyz = 6 defines z as an implicit function of x and y. Use implicit differentiation to compute the partial derivatives ∂z/∂x and ∂z/∂y at the point (1, 1, 1). 2) Find the local minimum, maximum, and saddle points – if any – of the function f(x, y) = 2x^2 + y^2 + 2x^2 y. 3) Find the maximum and minimum values of the function f(x, y) = xy on the closed and bounded region defined as {(x, y) | x^2 + y^2 ≤ 1}. [Hint: You need to analyze the function first on the open region {(x, y) | x^2 + y^2 < 1} and then on the boundary {(x, y) | x^2 + y^2 = 1}.]f 1 (x) = x and f2 (x) = sin (x) sin Wronskian functions that are linearly independent show using.