If the function $f(x)= a log x + \frac{b}{x}+x$ has extreme values at $ x= 1 $ and $x= 3 $, then (a, b) is :

$(-4, -3)$

The correct answer is Option (4) → $(-4, -3)$

$f(x)=a\log x+bx+x$

$f'(x)=\frac{a}{x}+b+1$

Since $f(x)$ has extreme value at $x=1$ and $x=3$, we set $f'(x)=0$

$\frac{a}{1}+b+1=0$

$a+b=-1$   $(x=1)$   ....(1)

$\frac{a}{3}+b+1=0$

$\frac{a}{3}+b=-1$   $(x=3)$   ....(2)

from (1) and (2),

$b=-1$ and $a=0$