Question

If $x^2{P}^{\prime }\left(x\right)+2xP\left(x\right)>0$ for all $x>0,P\left(2\right)=0,$ then which of the following is correct

• A. $P\left(x\right)\ge 0$ for $x>2$
• B. $P\left(x\right)>0$ for $x>2$
• C. $P\left(x\right)<0$ for $x>2$
• D. $P\left(x\right)\le 0$ for $x>2$

Answer 1

The correct option is B $P\left(x\right)>0$ for $x>2$

Given : $x^2{P}^{\prime }\left(x\right)+2xP\left(x\right)>0$
$\Rightarrow \frac{d\left(x^{2}P\right(x\left)\right)}{dx}>0$
So, curve $y=f\left(x\right)=x^{2}P\left(x\right)$ is increasing for all $x>0$
Now for $x>2$
$\Rightarrow f\left(x\right)>f\left(2\right)$
$\Rightarrow x^{2}P\left(x\right)>4\cdot P\left(2\right)$
$\Rightarrow P\left(x\right)>0$