Question

Let $h\left(x\right)=f\left(x\right)-{\left(f\left(x\right)\right)}^2+{\left(f\left(x\right)\right)}^3$ for every real number $x,$ then

• A. $h$ is increasing whenever $f$ is increasing
• B. $h$ is increasing whenever $f$ is decreasing
• C. $h$ is decreasing whenever $f$ is decreasing
• D. nothing can be said in general

Answer 1

Answer: (A), (C)

$h$ is increasing whenever $f$ is increasing
$h$ is decreasing whenever $f$ is decreasing

Here $h\left(x\right)=f\left(x\right)-{\left(f\left(x\right)\right)}^2+{\left(f\left(x\right)\right)}^3$

$\Rightarrow h^{\prime }\left(x\right)=f^{\prime }\left(x\right)-2f\left(x\right)f^{\prime }\left(x\right)+3{\left(f\left(x\right)\right)}^2{f}^{\prime }\left(x\right)$

$=f^{\prime }\left(x\right)(1-2f\left(x\right)+3{\left(f\left(x\right)\right)}^2)$

$=f^{\prime }\left(x\right)(3y^2-2y+1)$ where $y=f\left(x\right)$

The discriminant of $3y^2-2y+1$

$=4-12=-8<0$

and so its sign is the same as the coefficients of $y^2$ i.e., $3y^2-2y+1\forall y\in R$

$\therefore h^{\prime }\left(x\right)=f^{\prime }\left(x\right)(a$ positive quantity $)$

$\Rightarrow$ sign of $h^{\prime }\left(x\right)$ is the same as that of $f^{\prime }\left(x\right)$

$\Rightarrow$ either $h^{\prime }\left(x\right)>0$ and $f^{\prime }\left(x\right)>0$ or $h^{\prime }\left(x\right)<0$ and $f^{\prime }\left(x\right)<0$.

Hence $h\left(x\right)$ and $f\left(x\right)$ increases and decreases together.