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 $h(x)=f^2(x)+\left(f^{\prime}(x)\right)^2$ $\left(A^*\right)$ is monotonic increasing in $(-2,4)$ $\left( B ^*\right)$ has exactly 3 point of inflection. $\left(C^*\right)$ has exactly two points local maxima. $\left(D^*\right)$ 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) =2 f(x) f^{\prime}(x)+2 f^{\prime}(x) f^{\prime \prime}(x) $
$ =2 f^{\prime}(x)\left(f(x)+f^{\prime \prime}(x)\right)$
$ =2 f^{\prime}(x)\left(-g(x) f^{\prime}(x)\right)$
$=-2 g(x)\left(f^{\prime}(x)\right)^2$
$g(x) =(x+6)^{2009}(x+2)^{2011}(x+1)^{2010}(x-4)^{2013}(x-3)^{2012}(x-5)^{2014}$
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.]