Question

If $f\left(x\right)$ is a differentiable function and $f\left(x\right)-x^3$ is strictly increasing, then $f\left(x\right)+x-x^2$ is

• A. increasing
• B. decreasing
• C. not monotonic
• D. not depends on nature of $f\left(x\right)$

Answer 1

The correct option is A increasing

Let $H\left(x\right)=f\left(x\right)-x^3$
$\Rightarrow H^{\prime }\left(x\right)=f^{\prime }\left(x\right)-3x^2>0$
$\Rightarrow f^{\prime }\left(x\right)>3x^2$
Let $h\left(x\right)=f\left(x\right)+x-x^2$
$h^{\prime }\left(x\right)=f^{\prime }\left(x\right)+1-2x$
$\Rightarrow h^{\prime }\left(x\right)>3x^2-2x+1>0$
Hence, h(x) is increasing.