The function g(x)=f(x)/x where x not equal to 0 has an extreme value

Question :: The function g(x)=f(x)/x where x not equal to 0 has an extreme value, when

1. \[f^\prime\left(x\right)=f\left(x\right)\]

2. \[g^\prime\left(x\right)=f\left(x\right)\]

3.\[f\left(x\right)=0\]

4. \[g\left(x\right)=f^\prime\left(x\right)\]

Solution :

Given Function \[g\left(x\right)=\frac{f\left(x\right)}{x},\ x\neq0\] has an extreme value if \[g^\prime\left(x\right)=0\] \[\frac{xf^\prime\left(x\right)-f(x)}{x^2}=0\] \[xf^\prime\left(x\right)=f\left(x\right)\] \[f^\prime\left(x\right)=\frac{f\left(x\right)}{x}=g\left(x\right)\] Thus correct option is (4)