For the function of two variables f (x, y) = 4x3 + 6x2y2 − 7xy3 − 8y4 find the second order partial derivatives fxx, fyy and fxy (note that fxy should equal fyx).
Question
For the function of two variables f (x, y) = 4x3 + 6x2y2 − 7xy3 − 8y4 find the second order partial derivatives fxx, fyy and fxy (note that fxy should equal fyx).
Solution
Sure, let's find the second order partial derivatives for the given function.
The function is f(x, y) = 4x³ + 6x²y² - 7xy³ - 8y⁴.
First, let's find the first order partial derivatives:
fx = ∂f/∂x = 12x² + 12xy² - 7y³ fy = ∂f/∂y = 12x²y - 21xy² - 32y³
Now, let's find the second order partial derivatives:
fxx = ∂²f/∂x² = 24x + 12y² fyy = ∂²f/∂y² = 12x² - 42xy - 96y² fxy = ∂²f/∂x∂y = 24xy - 21y²
As per the Schwarz's theorem, the mixed partial derivatives fxy and fyx should be equal. So, let's find fyx:
fyx = ∂²f/∂y∂x = 24xy - 21y²
As expected, fxy = fyx.
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