Consider the function \( z = f(x, y) = x^3 + 6xy^2 \).

Find the equation for the tangent plane to \( z = f(x, y) \) at the point (3, 5).

Choose the correct equation from the options below:

A) \( z = 3x + 5y + 94 \)

B) \( z = 27x + 30y + 94 \)

C) \( z = 27x + 30y + 94xy \)

D) \( z = 27x + 30y + 94x^3 + 6xy^2 \)

The correct equation for the tangent plane to the function z = f(x, y) = [tex]x^3 + 6xy^2[/tex] at the point (3, 5) is z = 27x + 30y + 94, following the formula for a tangent plane at a point on a surface. The correct answer is option (B).

Explanation:

To find the equation of the tangent plane to the surface z = f(x, y) = [tex]x^3 + 6xy^2[/tex] at a point, we use the formula z = z_0 + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0), where f_x and f_y are partial derivatives of f with respect to x and y, respectively, and (x_0, y_0, z_0) is the point of tangency. First, calculate f_x = [tex]3x^2 + 6y^2[/tex] and f_y = 12xy. At the point (3, 5), f_x(3, 5) = 27 and f_y(3, 5) = 30. The value of the function at this point is z_0 = f(3, 5) = [tex]3^3 + 6(3)(5)^2[/tex] = 27 + 450 = 477. Thus, the equation of the tangent plane is z = 477 + 27(x - 3) + 30(y - 5), which simplifies to z = 27x + 30y + 94.