<s>[INST] Prove that $a^{4b}+b^{4a}\geq \frac{1}{2}$ Inspired by a problem of Vasile Cirtoaje I propose this :

Let $a,b>0$ such that $a+b=1$ then we have :
  $$a^{4b}+b^{4a}\geq \frac{1}{2}$$

I compute the derivative of $f(x)=x^{4(1-x)}+(1-x)^{4x}$ on $]0,1]$ we get :
$$ f'(x)=x^{4 (1 - x)} (\frac{4 (1 - x)}{x} - 4 \log(x)) + (1 - x)^{(4 x)} (4 \log(1 - x) - \frac{4 x}{1 - x})$$
If we denote by $g(x)$ the function :
$$g(x)=x^{4 (1 - x)} (\frac{4 (1 - x)}{x} - 4 \log(x))$$
We can rewrite the derivative as :
$$f'(x)=g(x)-g(1-x)$$
So it's remains to show that $g(x)\geq g(1-x)$ or $g(x)\leq g(1-x)$ 
So it remains to show that $g(x)$ is increasing or decreasing .
After that I'm stuck...
Any helps are very appreciated !
Thanks a lot for your time . 
 [/INST] Alternative Proof:
We need to prove that, for all $a$ in $(0, 1/2]$,
$$a^{4 - 4a} + (1 - a)^{4a} \ge 1/2.$$
We split into two cases:

Case I $\,\, a\in (0, 1/4)$:
We have
$$a^{4 - 4a} + (1 - a)^{4a}
\ge (1 - a)^{4a} \ge 1 - a > \frac12.$$

Case II $\,\,a \in [1/4, 1/2]$:
First, using Bernoulli inequality, we have
\begin{align*}
 a^{4 - 4a} 
 &= 2^{-1}2^{-(2 - 4a)}(1 - (1 - 2a))^{3 - 4a} a\\
 &\ge 2^{-1}2^{-(2 - 4a)}[1 - (1 - 2a)(3 - 4a)] a\\
 &\ge 2^{-1}[1 - (2 - 4a)\ln 2]\,[1 - (1 - 2a)(3 - 4a)]a
\end{align*}
where we have used $2^{-(2 - 4a)} = \mathrm{e}^{-(2 - 4a)\ln 2}
\ge 1 - (2 - 4a)\ln 2$.
Second, using Bernoulli inequality, we have
\begin{align*}
 (1 - a)^{4a} &= 2^{-2}2^{2 - 4a}(1 + (1 - 2a))^{4a}\\
 &\ge 2^{-2}2^{2 - 4a}[1 + (1 - 2a)\cdot 4a]\\
 &\ge 2^{-2}[1 + (2 - 4a)\ln 2]\,[1 + (1 - 2a)\cdot 4a]
\end{align*}
where we have used $2^{2 - 4a} = \mathrm{e}^{(2 - 4a)\ln 2}
\ge 1 + (2 - 4a)\ln 2$.
It suffices to prove that
\begin{align*}
 &2^{-1}[1 - (2 - 4a)\ln 2]\,[1 - (1 - 2a)(3 - 4a)]a\\
 &\quad + 2^{-2}[1 + (2 - 4a)\ln 2]\,[1 + (1 - 2a)\cdot 4a] \ge \frac12
\end{align*}
or
$$ (1 - 2a)^2[-16(\ln 2)a^2 + (20\ln 2 - 4)a + 2\ln 2 - 1]
\ge 0$$
which is true.

We are done.
</s>