TMUA Official 2022 Paper 2 · Question 20

The functions $f_1$ to $f_5$ are defined on the real numbers by

\begin{aligned} f_1(x)&=\cos x\\ f_2(x)&=\sin(\cos x)\\ f_3(x)&=\cos(\sin(\cos x))\\ f_4(x)&=\sin(\cos(\sin(\cos x)))\\ f_5(x)&=\cos(\sin(\cos(\sin(\cos x)))) \end{aligned}

where all numbers are taken to be in radians.

These functions have maximum values $m_1$ , $m_2$ , $m_3$ , $m_4$ and $m_5$ , respectively.

Which one of the following statements is true?

m_1

m_2

m_3

m_4

and

m_5

are all equal to 1

0<m_5<m_4<m_3<m_2<m_1=1

m_1=m_3=m_5=1

and

0<m_2=m_4<1

m_1=m_3=m_5=1

and

0<m_4<m_2<1

m_1=m_3=1

and

0<m_2=m_4<1

and

0<m_5<1

m_1=m_3=1

and

0<m_4<m_2<1

and

0<m_5<1

examday