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CMI-2018-DataScience-A: 16

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Let $a,b$ be numbers between $1$ and $2$ and let $c,d$ be numbers between $3$ and $4$. Let $u=a^{-1},v=b^{-1},w=c^{-1}$ and $x=d^{-1}$. Say which of the following inequalities are true:

  1. $(a+b+c+d)(u+v+w+x)>16$
  2. $(a^4+b^4+c^4+d^4)\leq 4abcd$
  3. $(a^2+b^2)wx\leq (c^2+d^2)uv$
  4. $d(a^3+b^3+c^3)<3abc$
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By A.M-G.M inequality we know that $\frac{a_1+a_2+\dots +a_n}{n}\geq \sqrt[n]{a_1 a_2 \dots a_n}$

Therefore we have $\frac{a+b+c+d}{4}> \sqrt[4]{abcd}$   and  $\frac{u+v+w+x}{4}> \sqrt[4]{uvwx}$.

[ Note: since a,b,c,d are not all equal therefore equality does not hold in this case]

Multiplying these inequalities $(a+b+c+d)(u+v+w+x)>16\sqrt[4]{abcduvwx}=16$

Therefore we got option A is true
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