Final answer:
To find the first and second order partial derivatives of the given function, differentiate it with respect to x and y. The first order partial derivatives are 3y² + 10xy and 6xy - 2 + 5x². The second order partial derivatives are 10y, 6x, and 6y + 10x. To find fxy(-1, 2), substitute x = -1 and y = 2 into the expression for fxy and evaluate.
Step-by-step explanation:
To find the first and second order partial derivatives for f(x, y) = 3xy² - 2y + 5x²y, we need to differentiate the function with respect to x and y.
The first order partial derivatives are:
fx = 3y² + 10xy
fy = 6xy - 2 + 5x²
The second order partial derivatives are:
f_xx = 10y
fyy = 6x
fxy = 6y + 10x
Finally, to determine the value of fxy(-1, 2), substitute x = -1 and y = 2 into the expression for fxy and evaluate:
fxy(-1, 2) = 6(2) + 10(-1) = 12 - 10 = 2