I.S.I. 10+2 Subjectives Solution

P148. Show that there is no real constant c > 0 such that \((\cos\sqrt{x+c}=\cos\sqrt{x})\) for all real numbers \((x\ge 0)\).
Solution:

If the given equation holds for some constant c>0 then,

f(x) = \((\cos\sqrt{x}-\cos\sqrt{x+c}=0)\) for all \((x\ge 0)\)
\((\Rightarrow 2\sin\frac{\sqrt{x+c}+\sqrt{x}}{2}\sin\frac{\sqrt{x+c}-\sqrt{x}}{2}=0)\)
Putting x=0, we note
\((\Rightarrow\sin^2\frac{\sqrt{c}}{2}=0)\)
As \((c\not=0)\)
\((\sqrt{c}=2n\pi)\)
\((\Rightarrow c=4n^2\pi^2)\)
We put n=1 and x=\((\frac{\pi}{2})\) to note that f(x) is not zero.
Hence no c>0 allows f(x) =0 for all \((x\ge 0)\). (proved)

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