Question

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

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

Answer 1

Answer: (A), (C)

'h' is decreasing wherever 'f' is decreasing
'h' is increasing whenever 'f' is increasing

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