Final Answer:
The equation of the normal line to f(x, y) at the point y = 0.5 is given by (c) 4x + 3y = 18.05.
Step-by-step explanation:
To find the equation of the normal line, we need to determine the gradient of the tangent at the given point. The gradient m of the tangent to the curve f(x, y) is found by differentiating the function f(x, y) with respect to x and then substituting the values of x and y at the given point. In this case, f(x, y) = 4y³ - kx + cos(y⁵ - y³), and at y = 0.5, we evaluate the gradient.
Next, the negative reciprocal of the gradient gives the slope of the normal line. With this slope and the given point (x, y) = (0.5, f(0.5)), we use the point-slope form of a line (y - y₁ = m(x - x₁)) to find the equation of the normal line.
By solving for y, we obtain the equation of the normal line. Comparing with the provided options, we find that (c) 4x + 3y = 18.05 is the correct equation.
In conclusion, finding the equation of the normal line involves calculating the gradient at the given point, determining the negative reciprocal as the slope of the normal line, and using the point-slope form to derive the equation. The correct option (c) accurately represents the equation of the normal line to f(x, y) at the specified point.