1. The problem is to understand how the gradients \(\nabla f = (1,1)\) and \(\nabla g = (2x, 2y)\) were calculated. 2. The gradient of a function \(f(x,y)\) is a vector of its partial derivatives: \(\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\). 3. For \(f(x,y) = x + y\), the partial derivatives are: \(\frac{\partial f}{\partial x} = 1\) because the derivative of \(x\) with respect to \(x\) is 1 and \(y\) is treated as a constant. \(\frac{\partial f}{\partial y} = 1\) because the derivative of \(y\) with respect to \(y\) is 1 and \(x\) is treated as a constant. So, \(\nabla f = (1,1)\). 4. For \(g(x,y) = x^2 + y^2\), the partial derivatives are: \(\frac{\partial g}{\partial x} = 2x\) by the power rule. \(\frac{\partial g}{\partial y} = 2y\) similarly. So, \(\nabla g = (2x, 2y)\). 5. These gradients represent the direction and rate of fastest increase of the functions \(f\) and \(g\) at any point \((x,y)\).