Question

Given that \(f: S \rightarrow R\) is said to have a fixed point at \(c\) of \(S\) if \(f(c)=c\).
Let \(f:[1, \infty) \rightarrow R\) be defined by \(f(x)=1+\sqrt{x}\). Then
(A) \(f\) has no fixed point in \([1, \infty)\)
(B) \(f\) has unique fixed point in \([1, \infty)\)
(C) \(f\) has two fixed points in \([1, \infty)\)
(D) \(f\) has infinitely many fixed points in \([1, \infty)\)

Answer 1

Ans: (B)
Hint:
$$
\begin{aligned}
&f(c)=c \\
&\Rightarrow c^{2}-3 c+1=0 \\
&\Rightarrow c=\frac{3 \pm \sqrt{5}}{2}
\end{aligned}
$$

Answer 2

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