Show with Lmvt1+2lnx<=x^(2)

Question

Hello Guys ,can u help me

I don't know how prove this probleme

Show that:::::

1+2lnx<=x^(2)

Answer 1

\(\because\) x² & exponential functions both are increasing functions

Also

ex² \(\leq\) ex²            \(\forall\) x \(\in\) (0, \(\infty\))

⇒ ln(ex²) \(\leq\) ln ex² (\(\because \) ln x is strictly increasing function)

⇒ ln e + ln x² \(\leq\) x² ln e (\(\because\) ln (AB) = ln A + ln B)

⇒ 1 + 2 ln x \(\leq\) x² (\(\because \) ln e = 1)