Final answer:
The gradient vector of the function f(x,y) at the point (2,1) is (0, 4), derived from calculating the partial derivatives with respect to x and y.
Step-by-step explanation:
The student is asking to find the gradient vector of the function f(x,y)=x² lny at the point (2,1). To do this, we need to calculate the partial derivatives of the function with respect to both x and y. The gradient vector is then composed of these partial derivatives.
The partial derivative of the function with respect to x is 2x lny, and at the point (2,1) it is 0 because lny is zero when y=1. The partial derivative with respect to y is x²/y, and at the point (2,1) it is 4. Therefore, the gradient vector at the point (2,1) is (0, 4).