Question

Assertion (A): If $f(x,y,z)=\sqrt{yz}+\sqrt{zx}+\sqrt{xy}$, then $x{f}_x+y{f}_y+z{f}_z=f(x,y,z)$

Reason (R): If $F\left(u\right)=f(x,y,z)$ is a homogeneous function of degree $1$ in $x,y,z$ then $x{u}_x+y{u}_y+z{u}_z=\frac{n.F\left(u\right)}{F^{\prime }\left(u\right)}$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

Assertion :

$f(x,y,z)=\sqrt{yz}+\sqrt{zx}+\sqrt{xy}$

$\frac{\partial f}{\partial x}=\frac{1}{2}\sqrt{\frac{z}{x}}+\frac{1}{2}\sqrt{\frac{y}{x}}$

$\frac{\partial f}{\partial y}=\frac{1}{2}\left(\sqrt{\frac{z}{y}}\right)+\frac{1}{2}\left(\sqrt{\frac{x}{y}}\right)$

$\frac{\partial f}{\partial z}=\frac{1}{2}(\sqrt{\frac{y}{z}}+\sqrt{\frac{x}{z}})$

$x{f}_x+y{f}_y+z{f}_z=\frac{1}{2}(\sqrt{zx}+\sqrt{yx}+\sqrt{zy}+\sqrt{xy}+\sqrt{yz}+\sqrt{xz})$ $=f(x,y,z)$

Reason

$\sum x\frac{\partial f\left(v\right)}{\partial x}=nF\left(u\right)$

$F{}^{\prime }(\sum x\frac{\partial y}{\partial x})=nF\left(u\right)$

$(\sum x\frac{\partial y}{\partial x})=\frac{nF\left(u\right)}{F{}^{\prime }\left(u\right)}$