A) \[{{a}^{2}}+{{c}^{2}}>{{b}^{2}}\]
B) \[{{a}^{2}}+{{b}^{2}}>2{{c}^{2}}\]
C) \[{{a}^{2}}+{{c}^{2}}>2{{b}^{2}}\]
D) \[{{a}^{2}}+{{b}^{2}}>{{c}^{2}}\]
Correct Answer: C
Solution :
From the section of inequality, we know that A.M. of \[{{n}^{th}}\] powers \[(n-1)=1,\ 2\ i.e.,\ n=2,\ 3\] power of A.M. \[i.e.\] \[\frac{1}{2}({{a}^{n}}+{{c}^{n}})>{{\left( \frac{1}{2}(a+c) \right)}^{n}}\] Considering two quantities \[{{b}^{2}}=ac\] and c or \[\frac{1}{2}({{a}^{n}}+{{c}^{n}})>{{(A)}^{n}}\] or \[\frac{1}{2}({{a}^{n}}+{{c}^{n}})>{{(H)}^{n}}\] Since A.M. >H.M. \[\frac{1}{2}({{a}^{n}}+{{c}^{n}})>{{(b)}^{n}}\] \[\Rightarrow \] \[{{a}^{n}}+{{c}^{n}}>2{{b}^{n}}\] Putting\[n=2\], we have\[{{a}^{2}}+{{c}^{2}}>2{{b}^{2}}\].
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