Question

If $f\left(x\right)=\frac{ax+b}{(x-1)(x-4)}$ has local maximum point at $(2,-1),$ then $(a+b)$ is equal to

Answer 1

$f\left(x\right)=\frac{ax+b}{(x-1)(x-4)}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{(x^2-5x+4)a-(ax+b)(2x-5)}{(x^2-5x+4{)}^2}$
As $x=2$ is a maxima ,
$\Rightarrow f^{\prime }\left(2\right)=0$
$\Rightarrow (2^2-5\times 2+4)a-(2a+b)(2\times 2-5)=0$
$\Rightarrow -2a+2a+b=0$
$\Rightarrow b=0$
And $f\left(2\right)=-1$
$f\left(2\right)=\frac{2a+b}{(2-1)(2-4)}$
$\Rightarrow -1=\frac{2a+b}{-2}$
$\Rightarrow 2a+b=2$
$\Rightarrow a=1$