Question

Let $F\left(x\right)=1+f\left(x\right)+{\left(f\right(x\left)\right)}^2+{\left(f\right(x\left)\right)}^3,$ where $f\left(x\right)$ is an increasing differentiable function and $F\left(x\right)=0$ has a positive root, then

• A. $F\left(x\right)$ is an increasing function
• B. $F\left(0\right)\le 0$
• C. $f\left(0\right)\le -1$
• D. $F^{\prime }\left(0\right)\ge 0$

Answer 1

Answer: (A), (B), (C), (D)

$F^{\prime }\left(0\right)\ge 0$

Given, $F\left(x\right)=1+f\left(x\right)+{\left(f\right(x\left)\right)}^2+{\left(f\right(x\left)\right)}^3$
$\Rightarrow F^{\prime }\left(x\right)=(1+2f(x)+3{\left(f\right(x\left)\right)}^2)f^{\prime }\left(x\right)\ge 0,$
$(\because 1+2f(x)+3f(x{)}^2>0,\forall x,\text{as}D<0)$
So, $F\left(x\right)$ is increasing.
So, $F\left(0\right)\le 0$ ($\therefore F\left(x\right)$ has a positive root & increasing both)
$\Rightarrow (1+f(0\left)\right)(1+(f\left(0\right){)}^2)\le 0$
$\Rightarrow f\left(0\right)\le -1$