2
$\begingroup$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ones also seemed easy after I grasped them, so I'm sure that it's just a bit hard for me to get the first thought on the right path...

$$x\in\mathbb{R}, x\geq 0$$ $$1+x^3\geq x+x^2$$

$\endgroup$
3
$\begingroup$

$$\begin{align} 1+x^3\geq x+x^2 &\iff x^3-x \geq x^2 - 1\\ \\ & \iff x(x^2-1) \geq x^2 - 1 \quad\forall x\geq 0\end{align}$$

$\endgroup$
4
$\begingroup$

we have to prove that
$1+x^3\geq x(1+x)$ for all $x$ with $x>0$, then we have $1+x^3-(1+x)x\geq 0$
$(1+x)(1+x^2-x)-x(1+x)\geq 0$ this is equivalent to
$(1+x)(1+x^2-2x)\geq 0$ or $(x-1)^2(x+1)\geq 0$ which is true. Sonnhard.

$\endgroup$
3
$\begingroup$

Hint: Both sides are multiples of $1+x$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.