Let \(f: R \rightarrow R\) be a bijection. \(A\) curve represented by \(y=f(x)\) is such that \(f^{\prime}(x)>0 \forall x \in \mathbf{R}\). The tangent and normal drawn at \(P(\alpha, 1)\) on the curve cuts the \(X\)-axis at \(A, B\) respectively and \(C\) is the foot of the perpendicular from \(P\) onto the \(X\)-axis. If \(P(\alpha, \mathrm{l})\) is such a point that \(A C+C B\) is minimum, then the tangent at \(P\) is parallel to the line

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.