Question

For a scalar function $f(x,y,z)=x^2+3y^2+2z^2,$ the gradient at the point $P(1,2,-1)$ is

• A. $2\hat{i}+6\hat{j}+4\hat{k}$
• B. $2\hat{i}+12\hat{j}-4\hat{k}$
• C. $2\hat{i}+12\hat{j}+4\hat{k}$
• D. $\sqrt{56}$

Answer 1

The correct option is B $2\hat{i}+12\hat{j}-4\hat{k}$

Given scalar function,
$f(x,y,z)=x^2+3y^2+2z^2$
$gradf=▽f$
$=\hat{i}\frac{\partial f}{\partial x}+\hat{j}\frac{\partial f}{\partial y}+\hat{k}\frac{\partial f}{\partial z}$
$=\hat{i}\left(2x\right)+\hat{j}\left(6y\right)+\hat{k}\left(4z\right)$
$gradf$ at $(1,2,-1)$
$=2\hat{i}+12\hat{j}-4\hat{k}$