Final answer:
The value of (d²y/dx²) at the point (4,3) is determined by differentiating the equation x² · y² = 25 twice with respect to x. The first differentiation, using the product rule, reveals that (dy/dx) = 0. Subsequently, the second derivative (d²y/dx²) is also 0, which corresponds to answer choice A.
Step-by-step explanation:
The question asks for the value of (d²y/dx²) at a given point when x² · y² = 25. To find this value, we need to differentiate the given equation with respect to x, twice. Starting by differentiating both sides with respect to x:
First differentiation (using the product rule):2xy(dy/dx) + 2x(dy/dx)y = 0.
Now, solve for dy/dx and then differentiate this result with respect to x to find d²y/dx².
At the point (4,3), we have:
2(4)(3)(dy/dx) + 2(4)(dy/dx)(3) = 0,
24(dy/dx) + 24(dy/dx) = 0,
48(dy/dx) = 0,
(dy/dx) = 0 (since 48 is not equal to zero).
Second differentiation:
Since (dy/dx) = 0, it implies that all terms in the second derivative that contain (dy/dx) will be zero, therefore (d²y/dx²) = 0 at the point (4,3).
The correct answer is A. 0.