In the interval [0, 1], the equation x = cos(x) has A. No solution B. Exactly one solution C. Exactly two solutions D. An infinite number of solutions

GATE 1995 · Engineering Mathematics · Maxima Minima · medium

Answer: B. Exactly one solution

1. Check endpoints of g(x) = x - cos(x): Sign change on [0,1]: by IVT at least one root exists.
2. Show g is strictly increasing: g is strictly increasing on [0,1], so it can cross zero at most once.
3. Conclude exactly one solution: The equation x = cos(x) has exactly one solution in [0,1], near x ≈ 0.739.