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Find the linear approximation of the function at the given point. \[ f(x, y)=-10 x^{2} y^{3} \text { at }(5,-2) \] A. \( L(x, y)=160 x-3,000 y-8,000 \) B. \( L(x, y)=800 x-600 y-8,000 \) C. \( L(x, y)

Answer :

Final answer:

The linear approximation of the function f(x, y)=-10x^2y^3 at the point (5, -2) is L(x, y) = 800x - 600y - 8000.

Explanation:

To find the linear approximation of the function at the given point, we need to use the formula for linear approximation. The linear approximation of a function f(x) at a point (a, b) is given by the equation L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b), where f_x and f_y are the partial derivatives of f with respect to x and y, respectively.

In this case, the function is f(x, y) = -10x^2y^3, and we want to find the linear approximation at the point (5, -2). The partial derivatives are f_x = -20xy^3 and f_y = -30x^2y^2. Evaluating these derivatives at the point (5, -2), we get f_x(5, -2) = -400 and f_y(5, -2) = -1500.

Plugging these values into the formula, we have L(x, y) = f(5, -2) + (-400)(x - 5) + (-1500)(y + 2). Simplifying this equation gives L(x, y) = -8000 + (-400)(x - 5) + (-1500)(y + 2). Therefore, the correct answer is B, L(x, y) = 800x - 600y - 8000.

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