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

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

Tardigrade Admin · Accepted Answer

f (2)=10, hence 2 ae -2 b =10 ⇒ ae -2 b =5 f prime(x)=a[e-b x-b x e-b x]=0 f prime(2)=0 a(e-2 b-2 b e-2 b)=0 a e-2 b(1-2 b)=0 ⇒ b=1 / 2 text or a=0 text (rejected) text from (1) if b=1 / 2 ; a=5 e ∴ a=5 e text and b=1 / 2

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

$a = 5; b = 0$

$a = 5 e, b = 1/2$

$a = 5 e^{2}, b = 1$

none

$f (2) = 10$ , hence $2 a e^{- 2 b} = 10 \Rightarrow a e^{- 2 b} = 5$
$f^{'} (x) = a [e^{- b x} - b x e^{- b x}] = 0$
$f^{'} (2) = 0$
$a (e^{- 2 b} - 2 b e^{- 2 b}) = 0$
$a e^{- 2 b} (1 - 2 b) = 0$
$\Rightarrow b = 1/2 or a = 0 (rejected)$
$from (1) if b = 1/2; a = 5 e$
$∴ a = 5 e and b = 1/2$

Q. If the function f(x)=axe−bx has a local maximum at the point (2,10) then

a=5;b=0

a=5e,b=1/2

a=5e2,b=1