Question

Consider the functions \(f_{1}(x)=x, f_{2}(x)=2+\log _{e} x, x>0\). The graphs of the functions intersect.
(A) once in \((0,1)\) but never in \((1, \infty)\)
(B) once in \((0,1)\) and once in \(\left(e^{2}, \infty\right)\)
(C) once in \((0,1)\) and once in \(\left(e, e^{2}\right)\)
(D) more than twice in \((0, \infty)\)

Answer 1

Ans: (C)
Hint: \(f_{1}(x)=x, f_{2}(x)=2+\log _{e} x\)
Let \(g(x)=f_{2}(x)-f_{1}(x)=2+\log _{e} x-x\)
\(g\left(0^{+}\right)<0, g(1)>0, g(e)>0, g\left(e^{2}\right)<0\)
and value of \(g(x)\) for all \(x \geq e^{2}\) is negative.
\(\therefore g(x)=0\) has two roots in \((0,1)\) and \(\left(e, e^{2}\right)\)

Answer 2

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Answer 3

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