WBJEE · Maths · Functions

Given that \(\mathrm{f}: \mathrm{S} \rightarrow \mathrm{R}\) is said to have a fixed point at \(\mathrm{c}\) of \(\mathrm{S}\) if \(\mathrm{f}(\mathrm{c})=\mathrm{c}\).
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 \([1, \infty)\)

Verified Solution

Answer & Solution

Correct Answer

(B) f has unique fixed point in \([1, \infty)\)

Step-by-step Solution

Detailed explanation

\(f(c)=c\) \(\Rightarrow c^{2}-3 c+1=0\) \(\Rightarrow c=\frac{3 \pm \sqrt{5}}{2}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Given that \(\mathrm{f}: \mathrm{S} \rightarrow \mathrm{R}\) is said to have a fixed point at \(\mathrm{c}\) of \(\mathrm{S}\) if \(\mathrm{f}(\mathrm{c})=\mathrm{c}\). Let \(f:[1, \infty) \rightarrow R\) be defined by \(f(x)=1+\sqrt{x}\). Then

Step-by-step Solution

Given that \(\mathrm{f}: \mathrm{S} \rightarrow \mathrm{R}\) is said to have a fixed point at \(\mathrm{c}\) of \(\mathrm{S}\) if \(\mathrm{f}(\mathrm{c})=\mathrm{c}\).
Let \(f:[1, \infty) \rightarrow R\) be defined by \(f(x)=1+\sqrt{x}\). Then