Which of the following is/are correct? Between any two roots of `e^xcosx=1,` there exists at least one root of `tanx=1.` Between any two roots of `e^xsinx=1,` there exists at least one root of `tanx=-1.` Between any two roots of `e^xcosx=1,` there exists at least one root of `e^xsinx=1.` Between any two roots of `e^xsinx=1,` there exists at least one root of `e^xcosx=1.`

Options

A. at least one root
B. at most one root
C. exuctly one root
D. no root

Answer

A

Explanation

Let `f(x)=e^(-x)-sin x ` and let `alpha and beta` be two roots of the equation `e^(x) sin x-1=0` such that `alpha lt beta`. Then,
`e^(alpha sin alpha=1 and e^(beta) sin beta=1`
`rArr e^(-alpha)-sin =0 and e^(-beta)-sin beta=0" "...(i)`
Clearly, f(x) is continuous on `[alpha, beta]` and differentiable on `(alpha, beta)`.
Also, `f(alpha)=f(beta)=0 " "`[ Uing (i)]
Therefore, by Rolle's theorem there exists ` c in (alpha,beta)` such that `f'(c)=0`
`rArr -e^(-c)- cos c=-0`
`rArr e^(-e) cos c+1=0`
`rArr x=c ` is a root of `e^(x) cos x+1=0`, where `c in (alpha, beta)`