Question

If $f\left(x\right)$ is a polynomial function and $f^{\prime }\left(x\right)>f\left(x\right),\forall x\ge 1$ and $f\left(1\right)=0$, then

• A. $f\left(x\right)\ge 0,\forall x\ge 1$
• B. $f\left(x\right)<0,\forall x\ge 1$
• C. $f\left(x\right)=0,\forall x\ge 1$
• D. None of the above.

Answer 1

The correct option is A $f\left(x\right)\ge 0,\forall x\ge 1$

$f^{\prime }\left(x\right)>f\left(x\right)\forall x>1\Rightarrow f^{\prime }\left(x\right)>f(x=1)=f\left(1\right)\Rightarrow f^{\prime }\left(x\right)>0$

$\Rightarrow f\left(x\right)$ is strictly increasing for $x>1$

But $f\left(1\right)=0$

$\Rightarrow f\left(x\right)>f\left(1\right)\forall x>1\Rightarrow f\left(x\right)>0\forall x>1f\left(1\right)=0$

Hence, $f\left(x\right)\ge 0\forall x\ge 1$