Answer: (a) Let and be continuous on the closed interval If and prove that there exists

Chapter 1, Problem 120

(choose chapter or problem)

(a) Let \(f_{1}(x)\) and \(f_{2}(x)\) be continuous on the closed interval [a, b]. If \(f_{1}(a)<f_{2}(a)\) and \(f_{1}(b)>f_{2}(b)\), prove that there exists c between a and b such that \(f_{1}(c)=f_{2}(c)\).

(b) Show that there exists c in \(\left[0, \frac{\pi}{2}\right]\) such that cos x = x. Use a graphing utility to approximate c to three decimal places.

Text Transcription:

f_1 (x)

f_2 (x)

f_1 (a) < f_2 (a)

f_1 (b) > f_2 (b)

f_1 (c) = f_2 (c)

[0, pi/2]