GATE 2023 EE Q.65: solution and answer

Official answer: 10Source: Official answer key, GATE 2023 EE (IIT Kanpur)

1. What is being asked. At a point, a function increases fastest in the direction of its gradient $\nabla f$, and the rate of change in that direction is $|\nabla f|$. So we need $|\nabla f(1,1)|$.

2. Partial derivative with respect to $x_1$: $\frac{\partial f}{\partial x_1}=2x_1+3+x_2$. At $(1,1)$ this is $2+3+1=6$.

3. Partial derivative with respect to $x_2$: $\frac{\partial f}{\partial x_2}=4x_2+3+x_1$. At $(1,1)$ this is $4+3+1=8$.

4. Gradient at the point: $\nabla f(1,1)=6\hat{i}+8\hat{j}$.

5. Its magnitude: $$|\nabla f(1,1)|=\sqrt{6^2+8^2}=\sqrt{100}=10$$

6. The maximum rate of change is exactly $10$, so rounding to the nearest integer leaves it at $10$.