Question

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

• A. is monotonic increasing in $(–2,4)$
• B. has exactly $3$ point of inflection
• C. has exactly two points local maxima
• D. has a negative point of local minima.

Answer 1

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

The correct options are
A is monotonic increasing in $(–2,4)$
B has exactly $3$ point of inflection
C has exactly two points local maxima
D has a negative point of local minima.

Given,
$h\left(x\right)=f^2\left(x\right)+\left(f^{{}^{\prime }}\right(x){)}^2$
differentiation with respect to $x$
$\Rightarrow h^{{}^{\prime }}\left(x\right)=2f\left(x\right)f^{{}^{\prime }}\left(x\right)+2f^{{}^{\prime }}\left(x\right)f^{{}^{\prime \prime }}\left(x\right)$
$\Rightarrow h^{{}^{\prime }}\left(x\right)=2f^{{}^{\prime }}\left(x\right)\left(f\right(x)+f^{{}^{\prime \prime }}(x\left)\right)$
$\Rightarrow h^{{}^{\prime }}\left(x\right)=2f^{{}^{\prime }}\left(x\right)(-g(x\left)f^{{}^{\prime }}\right(x\left)\right)$
$\Rightarrow h^{{}^{\prime }}\left(x\right)=-2g\left(x\right)\left(f^{{}^{\prime }}\right(x){)}^2$
$\Rightarrow h^{{}^{\prime }}\left(x\right)=-2\left(f^{{}^{\prime }}\right(x\left){)}^2\right(x+6{)}^{2009}(x+1{)}^{2010}(x+2{)}^{2011}(x–3{)}^{2012}(x–4{)}^{2013}(x–5{)}^{2014}$
Sign of $h^{{}^{\prime }}\left(x\right)$


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