Question

The magnitude of gradient for the scalar function $f(x,y,z)=x^2+\beta y^2+z^3$ at point (1, 1, 1) is 7 . The value of $|\beta |$ is _______.

• A. 3

Answer 1

The correct option is A 3
We Know that,

$▽f=\frac{\partial f}{\partial x}{\hat{a}}_x+\frac{\partial f}{\partial y}{\hat{a}}_y+\frac{\partial f}{\partial z}{\hat{a}}_z$

$=2x{\hat{a}}_x+2\beta y{\hat{a}}_y+3z^2{\hat{a}}_z$

$▽f_{(1,1,1)}=2{\hat{a}}_x+2\beta {\hat{a}}_y+3{\hat{a}}_z$

$|▽f|=\sqrt{4+4{\beta }^2+9}$

$(7{)}^2=13+4{\beta }^2$

$\beta =\pm 3$

$\beta$ = 3