The gradient of $f ( x , y , z ) = 2 x ^ { 3 } + x y ^ { 2 } + z ^ { 2 } x$ at (1, 1, 1) is

A) $- 8 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
B) $8 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k }$
C) $8 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$
D) $8 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
E) $8 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$

Question

The gradient of $f ( x , y , z ) = 2 x ^ { 3 } + x y ^ { 2 } + z ^ { 2 } x$ at (1, 1, 1) is

A) $- 8 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
B) $8 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k }$
C) $8 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$
D) $8 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
E) $8 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$

Quizplus · Accepted Answer

The gradient of a function $f(x, y, z)$ is given by $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$. For $f(x, y, z) = 2x^3 + xy^2 + z^2x$, we have $\frac{\partial f}{\partial x} = 6x^2 + y^2 + z^2$, $\frac{\partial f}{\partial y} = 2xy$, and $\frac{\partial f}{\partial z} = 2xz$. Substituting $x = 1$, $y = 1$, and $z = 1$, we get $\nabla f = (6 + 1 + 1)\mathbf{i} + (2 \cdot 1)\mathbf{j} + (2 \cdot 1)\mathbf{k} = 8\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$.

The Gradient Of f(x,y,z)=2x3+xy2+z2xf ( x , y , z ) = 2 x ^ { 3 } + x y ^ { 2 } + z ^ { 2 } xf(x,y,z)=2x3+xy2+z2x

The Gradient Of $f ( x , y , z ) = 2 x ^ { 3 } + x y ^ { 2 } + z ^ { 2 } x$