The functions f1 to f5 are defined on the real numbers by f1(x) = cos x f2(x) = sin(cos x) f3(x) = cos(sin(cos x)) f4(x) = sin(cos(sin(cos x))) f5(x) = cos(sin(cos(sin(cos x)))) where all numbers are taken to be in radians.
These functions have maximum values m1, m2, m3, m4 and m5 respectively.
Which one of the following statements is true?
- Am1, m2, m3, m4 and m5 are all equal to 1
- B0 < m5 < m4 < m3 < m2 < m1 = 1
- Cm1 = m3 = m5 = 1 and 0 < m2 = m4 < 1
- Dm1 = m3 = m5 = 1 and 0 < m4 < m2 < 1
- Em1 = m3 = 1 and 0 < m2 = m4 < 1 and 0 < m5 < 1
- Fm1 = m3 = 1 and 0 < m4 < m2 < 1 and 0 < m5 < 1
Show the answer and worked solution
answer · E
- Am1, m2, m3, m4 and m5 are all equal to 1
- B0 < m5 < m4 < m3 < m2 < m1 = 1
- Cm1 = m3 = m5 = 1 and 0 < m2 = m4 < 1
- Dm1 = m3 = m5 = 1 and 0 < m4 < m2 < 1
- Em1 = m3 = 1 and 0 < m2 = m4 < 1 and 0 < m5 < 1
- Fm1 = m3 = 1 and 0 < m4 < m2 < 1 and 0 < m5 < 1
Work outwards. m1 = 1, attained at x = 0. For f2, cos x ranges over [−1, 1] and sin increases there, so m2 = sin 1 ≈ 0.84. For f3, the inner value sin(cos x) can be made exactly 0 by taking x = π2, so m3 = cos 0 = 1. For f4, the inner cos(sin(cos x)) reaches 1, so m4 = sin 1 = m2. For f5, the innermost value cos(sin(cos x)) lies in [cos 1, 1], so sin of it lies in [sin(cos 1), sin 1], all strictly positive — and cos of a strictly positive number is strictly less than 1, so 0 < m5 < 1.