Answer :
Final answer:
To calculate du2f(2, 1), find the second-order partial derivatives of f(x, y) and evaluate them at the point (2, 1). Then, multiply the second-order partial derivatives by the components of u. The correct answer is approximately 5.40. Therefore, the correct option is d) 50.
Explanation:
To calculate du2f(2, 1), we need to find the second-order partial derivatives of f(x, y) and evaluate them at the point (2, 1). The second-order partial derivatives are found by taking the partial derivatives of the first-order partial derivatives.
First, let's find the first-order partial derivatives of f(x, y):
∂f/∂x = 3x^2 - 10xy
∂f/∂y = -5x^2 + 3y^2
Next, let's find the second-order partial derivatives:
∂^2f/∂x^2 = 6x - 10y
∂^2f/∂y^2 = 6y
Now, let's evaluate the second-order partial derivatives at the point (2, 1):
∂^2f/∂x^2(2, 1) = 6(2) - 10(1) = 2
∂^2f/∂y^2(2, 1) = 6(1) = 6
Finally, multiply the second-order partial derivatives by the components of u:
du2f(2, 1) = (2)(5/13)^2 + (6)(12/13)^2 = 50/169 + 864/169 = 914/169
So, the answer is approximately 5.40. Therefore, the correct option is d) 50.
