Question

Let \( f(x, y, z) = 4x^2 + 7xy + 3xz^2 \). The direction in which the function \( f(x, y, z) \) increases most rapidly at point \( P = (1, 0, 2) \) is

• A. ( 20\hat{i} + 7\hat{j} \)
• B. ( 20\hat{i} + 7\hat{j} + 12\hat{k} \)
• C. ( 20\hat{i} + 12\hat{k} \)
• D. ( 20\hat{i} \)

Hint:

The gradient vector \( \nabla f(x, y, z) \) points in the direction of the greatest rate of increase of the function, and its magnitude represents the rate of increase in that direction.

Answer 1

The Correct Option is B

Step 1: Gradient of the function.
The direction in which the function increases most rapidly is given by the gradient vector \( \nabla f(x, y, z) \). The gradient is the vector of partial derivatives of \( f \) with respect to \( x \), \( y \), and \( z \): \[ \nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right). \] Step 2: Calculating the partial derivatives.
For \( f(x, y, z) = 4x^2 + 7xy + 3xz^2 \), we calculate: \[ \frac{\partial f}{\partial x} = 8x + 7y + 3z^2, \quad \frac{\partial f}{\partial y} = 7x, \quad \frac{\partial f}{\partial z} = 6xz. \] Step 3: Evaluate the gradient at \( P = (1, 0, 2) \).
Substitute \( x = 1 \), \( y = 0 \), and \( z = 2 \) into the gradient components: \[ \frac{\partial f}{\partial x} = 8(1) + 7(0) + 3(2)^2 = 8 + 0 + 12 = 20, \frac{\partial f}{\partial y} = 7(1) = 7, \frac{\partial f}{\partial z} = 6(1)(2) = 12. \] Thus, the gradient is \( \nabla f(1, 0, 2) = 20\hat{i} + 7\hat{j} + 12\hat{k} \). Step 4: Conclusion.
The direction in which the function increases most rapidly at \( P \) is given by the vector \( 20\hat{i} + 7\hat{j} + 12\hat{k} \), so the correct answer is (B).