Question

Let $f(x)$ be a twice differentiable function and has no critical point and $g(x)=(x+6{)}^{2009}(x+1{)}^{2010}(x+2{)}^{2011}(x-3{)}^{2012}(x-4{)}^{2013}(x-5{)}^{2014}$ be such that $f(x)+g(x)f^{\prime }(x)+f^{\prime \prime }(x)=0$ then function $h(x)=f^2(x)+{\left(f^{\prime }(x)\right)}^2$ $\left(A^{\ast }\right)$ is monotonic increasing in $(-2,4)$ $\left(B^{\ast }\right)$ has exactly 3 point of inflection. $\left(C^{\ast }\right)$ has exactly two points local maxima. $\left(D^{\ast }\right)$ has a negative point of local minima.

• 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)

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

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