Question

A quadratic function of two variables is given as \[ f(x_1, x_2) = x_1^2 + 2x_2^2 + 3x_1 + 3x_2 + x_1x_2 + 1 \] The magnitude of the maximum rate of change of the function at the point (1,1) is ................... (Round off to the nearest integer).

Hint:

The maximum rate of change of a multivariable function at a point is equal to the magnitude of its gradient vector at that point.

Answer 1

Step 1: Gradient of function.
\[ \nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2} \right) \] \[ \frac{\partial f}{\partial x_1} = 2x_1 + x_2 + 3 \] \[ \frac{\partial f}{\partial x_2} = 4x_2 + x_1 + 3 \]

Step 2: Evaluate at (1,1).
\[ \frac{\partial f}{\partial x_1} = 2(1) + 1 + 3 = 6 \] \[ \frac{\partial f}{\partial x_2} = 4(1) + 1 + 3 = 8 \] So gradient = (6, 8).

Step 3: Maximum rate of change.
Magnitude of gradient = \[ |\nabla f| = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10 \] Correction: Exactly 10, not 9.

Final Answer:
\[ \boxed{10} \]