# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3$ is the minimal value for which the following inequality holds:
$$a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}$$

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27.4" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27.4" _i="1" _address="0.1.0.2.1" hover_enabled="0"]Choose specific values of $a,b,c,d$ to show that no value of $x$ less than 3 works. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27.4" _i="2" _address="0.1.0.2.2" hover_enabled="0"]Choosing $a=b=c=t$, the inequality becomes $3t^x+\frac{1}{t^{3x}}\ge \frac{3}{t}+t^3$. Note that, as $t\to\infty$, the LHS grows as $O(t^x)$ and the RHS grows as $O(t^3)$. For the LHS to be always greater than the RHS, it should be of a higher order. Hence, $x\ge 3$. Now show that the inequality is true for $x=3$. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27.4" _i="3" _address="0.1.0.2.3" hover_enabled="0"]Note that the RHS can be rewritten as $abc+bcd+cda+bda$. This reminds us of the AM-GM inequality. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27.4" _i="4" _address="0.1.0.2.4" hover_enabled="0"]Indeed, $abc+bcd+cda+bda\le \frac{a^3+b^3+c^3}{3}+\frac{d^3+b^3+c^3}{3}+\frac{a^3+d^3+c^3}{3}+\frac{a^3+b^3+d^3}{3}=a^3+b^3+c^3+d^3$. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]