Question

The magnitude of gradient for the scalar function $f(x,y,z)=x^3+\beta y^3+z^3$ at point (1, 1, 1) is 9. The magnitude of $\beta$ is_______.(Answer upto 2 decimal places)

• A. 2.65

Answer 1

The correct option is A 2.65
We know, $\nabla f=\frac{\partial f}{\partial x}\widehat{a_x}+\frac{\partial f}{\partial y}\widehat{a_y}+\frac{\partial f}{\partial z}\widehat{a_z}$

$=3x^2\widehat{a_x}+3\beta y^2\widehat{a_y}+3z^2\widehat{a_z}$

$\nabla f_{(1,1,1)}=3\widehat{a_x}+3\beta \widehat{a_y}+3\widehat{a_z}$

$\left|\nabla f_{(1,1,1)}\right|=\sqrt{9+9{\beta }^2+9}=9$

or, $18+9{\beta }^2=81$

or, $|\beta |=\sqrt{7}$