To find the gradient of the function f(x, y, z) = 3x²y - y³z², we need to find the partial derivatives with respect to each variable:
∂f/∂x = 6xy
∂f/∂y = 3x² - 3y²z²
∂f/∂z = -2y³z
So the gradient of f is given by the vector:
∇f(x, y, z) = <6xy, 3x² - 3y²z², -2y³z>
At the point (1, -2, -1), we have:
∇f(1, -2, -1) = <6(1)(-2), 3(1)² - 3(-2)²(-1)², -2(-2)³(-1)>
= <-12, -15, 16>
Therefore, the gradient of f at the point (1, -2, -1) is the vector <-12, -15, 16>.