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A good problem

Find no. of solutions of \(\large(x^2-7x+11)^{(x-6)(x-5)}=x\)

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Note by Mehul Chaturvedi
2 years ago

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Taking log both sides,

\((x-5)(x-6)\ln (x^2-7x+11)=\ln x\)

Note that \(x=5\) is a repeated root so the differential would be 0 at \(x=5\). Try to plot the graph. At \(x=\infty,\ LHS=\infty\). At \(x=5,6,2\), \(LHS=0\). At \(x=0,\ LHS>0\). I think graphs can be plotted now and it will intersect \(y=\ln x\) at two points. Pranjal Jain · 2 years ago

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@Mehul Chaturvedi I'll post more tomorrow, but for now, I'd just say to create a sign chart of where the exponent and where the \(x^2-7x+11\) are positive and when negative and where the LHS exceeds the RHS. The problem is that if the LHS intersects the RHS when negative, You have to watch for extraneous solutions because \((-2)^{\frac{7}{12}}\) is not an answer since its imaginary (the answer isn't 2^7/12, Im just using it as an example. Trevor Arashiro · 2 years ago

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