Question

Let $f\left(x\right)$ be a twice differentiable function and has no critical points. Let $g\left(x\right)=(x+6{)}^{2009}(x+1{)}^{2010}(x+2{)}^{2011}(x-3{)}^{2012}(x-4{)}^{2013}(x-5{)}^{2014}$ be such that $f\left(x\right)+g\left(x\right)f^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)=0.$ Then the function $h\left(x\right)=f^2\left(x\right)+(f^{\prime }\left(x\right){)}^2$

• A. is monotonically increasing in $(-2,4)$.
• B. has exactly three points of inflection.
• C. has exactly two points of local maxima.
• D. has a negative point of local minimum.

Answer 1

Answer: (A), (B), (C), (D)

has a negative point of local minimum.

$h\left(x\right)=f^2\left(x\right)+(f^{\prime }\left(x\right){)}^2$
$\Rightarrow h^{\prime }\left(x\right)=2f\left(x\right)f^{\prime }\left(x\right)+2f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)$
$=2f^{\prime }\left(x\right)[f\left(x\right)+f^{\prime \prime }\left(x\right)]$
$=2f^{\prime }\left(x\right)[-g\left(x\right)f^{\prime }\left(x\right)]$
$=-2g\left(x\right)(f^{\prime }\left(x\right){)}^2$

$g\left(x\right)=(x+6{)}^{2009}(x+1{)}^{2010}(x+2{)}^{2011}(x-3{)}^{2012}(x-4{)}^{2013}(x-5{)}^{2014}$

Sign of $h^{\prime }\left(x\right)$



$h^{\prime }\left(x\right)>0$ in $(-2,4)$
$\Rightarrow h\left(x\right)$ is monotonically increasing in $(-2,4)$
$x=-1,3$ and $5$ are points of inflection.
$x=-6$ and $x=4$ are points of local maxima.
$x=-2$ is only the point of local minimum.