<s>[INST] Prove that $4x-x^4 \leq 3, x \in \Bbb R$ How can I tackle the following inequality :  

Prove that $4x-x^4 \leq 3$, where $x$  is any real number. 

Can someone point me in the right direction?
 [/INST] We see that ,  $x^4-4x+3=x^4-2x^2+1+2x^2-4x+2=(x^2-1)^2+2(x-1)^2 \geq 0$ and so $-(x^4-4x+3)\leq 0 \implies  -x^4+4x-3 \leq 0$ and 
hence   $4x-x^4 \leq 3$
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