Question

If $h\left(x\right)=f\left(x\right)-{\left(f\left(x\right)\right)}^2+{\left(f\left(x\right)\right)}^3\forall xϵR$ then

• A. $h\left(x\right)$ increases when ever $f\left(x\right)$ decreases
• B. $h\left(x\right)$ increases whenever $f\left(x\right)$ increases
• C. $h\left(x\right)$ decreases whenever $f\left(x\right)$ decreases
• D. Nothing can be said regarding $h\left(x\right)$ & $f\left(x\right)$

Answer 1

Answer: (B), (C)

The correct options are
B $h\left(x\right)$ increases whenever $f\left(x\right)$ increases
C $h\left(x\right)$ decreases whenever $f\left(x\right)$ decreases

$h\left(x\right)=f\left(x\right)-\left(f\right(x){)}^2+(f\left(x\right){)}^3$
$h^{\prime }\left(x\right)=f^{\prime }\left(x\right)-2f\left(x\right).f^{\prime }\left(x\right)+3\left(f\right(x){)}^2.f^{\prime }(x)=f^{\prime }(x\left)\right(3\left(f\right(x){)}^2-2f(x)+1)$
Now discriminant of $3\left(f\right(x){)}^2-2f(x)+1$ is negative, therefore it will always positive
Hence sign of $h^{\prime }\left(x\right)$ is same as sign of $f^{\prime }\left(x\right)$
Thus if $f\left(x\right)$ is increasing then $h\left(x\right)$ will also increasing.
and if $f\left(x\right)$ is decreasing then $h\left(x\right)$ will also decreasing..