Question

If a continuous function $e^{-x}f\left(x\right)$ has only one stationary point and attains its maximum in $[1,3]$ at $x=2,$ then which of the following(s) is/are correct.

• A. $f\left(x\right)<f^{\prime }\left(x\right)$ for $1<x<2$
• B. $f\left(x\right)<f^{\prime }\left(x\right)$ for $2<x<3$
• C. $f\left(x\right)>f^{\prime }\left(x\right)$ for $1<x<2$
• D. $f\left(x\right)>f^{\prime }\left(x\right)$ for $2<x<3$

Answer 1

Answer: (A), (D)

$f\left(x\right)>f^{\prime }\left(x\right)$ for $2<x<3$

For a continuous function in $[1,3]$ and has a maxima at $x=2$
Let $g\left(x\right)=e^{-x}f\left(x\right)$
$\Rightarrow g^{\prime }\left(x\right)>0$ for $1<x<2$
$\Rightarrow e^{-x}\left(f^{\prime }\right(x)-f(x\left)\right)>0$ for $1<x<2$
$\Rightarrow f^{\prime }\left(x\right)>f\left(x\right)$ for $1<x<2$

$\Rightarrow g^{\prime }\left(x\right)<0$ for $2<x<3$
$\Rightarrow e^{-x}\left(f^{\prime }\right(x)-f(x\left)\right)<0$ for $2<x<3$
$\Rightarrow f^{\prime }\left(x\right)<f\left(x\right)$ for $2<x<3$