Question

Let $f:R\to R$ has a continuous third derivative such that $f\left(x\right)f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)f^{\prime \prime \prime }\left(x\right)\ne 0$ for any real $x$ and $f\left(x\right)f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)f^{\prime \prime \prime }\left(x\right)=g\left(x\right)$ Then

• A. $\underset{x\to \infty }{lim}g\left(x\right)$ can be zero
• B. $\underset{x\to \infty }{lim}g\left(x\right)$ can be negative
• C. $\underset{x\to \infty }{lim}g\left(x\right)$ can be $5$
• D. $\underset{x\to \infty }{lim}g\left(x\right)$ can be $(-e)$

Answer 1

Answer: (A), (C)

The correct options are
A $\underset{x\to \infty }{lim}g\left(x\right)$ can be zero
C $\underset{x\to \infty }{lim}g\left(x\right)$ can be $5$

Assume $f^{\prime \prime }\left(x\right)>0$ $($else replace $f\left(x\right)$ by $-f\left(x\right))$

Also assume $f^{\prime \prime \prime }\left(x\right)>0$ $($else replace $f\left(x\right)$ by $f(-x))$
It do not disturb sign of $g\left(x\right)$
Now $f^{\prime \prime }\left(x\right)>0$ $\Rightarrow$ $f^{\prime }\left(x\right)$ is increasing and $f^{\prime \prime \prime }\left(x\right)>0$
$\Rightarrow$ $f^{\prime }\left(x\right)$ is convex down (lies above tangent)
Hence $f^{\prime }(x+a)>f^{\prime }\left(x\right)+a{f}^{\prime \prime }\left(x\right)$ for all $x$ and $a$ so for a sufficiently large ${}^{\prime }{a}^{\prime }$ $f^{\prime }(x+a)$ is positive
$\Rightarrow$ $f^{\prime }\left(x\right)$ is always positive
Similarly $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ $\Rightarrow$ $f\left(x\right)$ is always positive
$\Rightarrow$ $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ $\Rightarrow$ $f\left(x\right)$ is always positive
$\Rightarrow$ $g\left(x\right)>0$ $\forall$ $x$ taking $f\left(x\right)=$ $e^x$ or $e^{-x}$ serves as an example of A or C