Question

The greatest rate of increase of \(f=xy^2z^3\) at the point \((0,-1,-2)\) is :

• A. \(4\)
• B. \(8\)
• C. \(10\sqrt{2}\)
• D. \(-8i\)

Hint:

The maximum directional derivative of a function is always equal to the magnitude of the gradient vector: \[ D_{\max}f=|\nabla f| \] The gradient direction always gives the direction of steepest increase.

Answer 1

The Correct Option is B

Concept: The greatest rate of increase of a scalar function occurs in the direction of the gradient vector. For a scalar function \(f(x,y,z)\), \[ \nabla f= \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \] The magnitude of the gradient gives the maximum directional derivative: \[ D_{\max}f=|\nabla f| \] Hence we first compute the gradient and then evaluate its magnitude at the given point.

Step 1: Writing the given function. We are given: \[ f=xy^2z^3 \] We now calculate all first-order partial derivatives carefully.

Step 2: Finding \( \dfrac{\partial f}{\partial x} \). Treating \(y\) and \(z\) as constants: \[ \frac{\partial f}{\partial x}=y^2z^3 \] Substitute: \[ y=-1,\quad z=-2 \] Then: \[ =(-1)^2(-2)^3 \] \[ =1(-8) \] \[ =-8 \]

Step 3: Finding \( \dfrac{\partial f}{\partial y} \). Differentiate with respect to \(y\): \[ \frac{\partial f}{\partial y}=2xyz^3 \] Substitute the point: \[ 2(0)(-1)(-8)=0 \] Thus: \[ \frac{\partial f}{\partial y}=0 \]

Step 4: Finding \( \dfrac{\partial f}{\partial z} \). Differentiate with respect to \(z\): \[ \frac{\partial f}{\partial z}=3xy^2z^2 \] Substitute the point: \[ 3(0)(1)(4)=0 \] Hence: \[ \frac{\partial f}{\partial z}=0 \]

Step 5: Constructing the gradient vector. Therefore: \[ \nabla f(0,-1,-2)=(-8,0,0) \]

Step 6: Finding the magnitude of the gradient. The magnitude is: \[ |\nabla f| = \sqrt{(-8)^2+0^2+0^2} \] \[ = \sqrt{64} \] \[ =8 \] Hence the greatest rate of increase is: \[ \boxed{8} \] Therefore the correct answer is: \[ \boxed{(2)\;8} \]