Question

If the function $f( x )= ax e ^{- bx }$ has a local maximum at the point $(2,10)$ then

• A. $a =5 ; b =0$
• B. $a =5 e , b =1 / 2$
• C. $a =5 e ^2, b =1$
• D. none

Answer 1

Answer: (B)

$f (2)=10$, hence $2 ae ^{-2 b }=10 \Rightarrow ae ^{-2 b }=5$
$f^{\prime}(x)=a\left[e^{-b x}-b x e^{-b x}\right]=0 $
$f^{\prime}(2)=0 $
$a\left(e^{-2 b}-2 b e^{-2 b}\right)=0$
$a e^{-2 b}(1-2 b)=0$
$ \Rightarrow b=1 / 2 \text { or } a=0 \text { (rejected) }$
$\text { from (1) if } b=1 / 2 ; a=5 e $
$\therefore a=5 e \text { and } b=1 / 2$