<s>[INST] Roots of the equation $(1-4x)^4+32x^4=\frac{1}{27}​$ find all real roots of the equation $(1-4x)^4+32x^4=\dfrac{1}{27}​$
i try to use binomial expansion and am-gm inequality but i don't know how to do next. 
 [/INST] Clearly there are no solutions if $ x > \frac{1}{4}$ or $ x < 0$. So, we may apply the power mean inequality to $ 1-4x, 2x, 2x$.
$$\sqrt[4]{\frac{( 1- 4x)^ 4 + (2x)^4 + (2x)^4 }{3}}\geq   \frac{ (1-4x) + 2x + 2x } { 3} =  \frac{1}{3} $$
This tells us that $( 1- 4x)^ 4 + 32x^4 \geq \frac{1}{27}$. 
Equality holds iff $ 1 -4x = 2x = 2x$, or that $ x = \frac{1}{6}$. 
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