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 :
Answer & explanation
Correct answer: option 4
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 $a=0$