If α, β ∈((π/2), π) and α

Question

If α, β ∈((π/2), π) and α<β, then which one of the following is true? (A) e cos α- cos β<(α/β) (B) e cos β- cos α<(β/α) (C) e cos α- cos β<(β/α) (D) e cos β- cos α<(α/β)

Tardigrade Admin · Accepted Answer

Consider f(x)=(e cos x/x) Now, f prime(x)=(-e cos x(1+x sin x)/x2) As f prime(x)<0 ⇒ f(α)>f(β) So, ( e cos α/α)>( e cos β/β) ⇒ e cos α- cos β>(α/β).

Q. If $α, β \in (\frac{π}{2}, π)$ and $α < β$ , then which one of the following is true?

$e^{c o s α - c o s β} < \frac{α}{β}$

$e^{c o s β - c o s α} < \frac{β}{α}$

$e^{c o s α - c o s β} < \frac{β}{α}$

$e^{c o s β - c o s α} < \frac{α}{β}$

Consider $f (x) = \frac{e ^{c o s x}}{x}$
Now, $f^{'} (x) = \frac{- e ^{c o s x} ( 1 + x s i n x )}{x ^{2}}$
As $f^{'} (x) < 0 \Rightarrow f (α) > f (β)$
So, $\frac{e ^{c o s α}}{α} > \frac{e ^{c o s β}}{β} \Rightarrow e^{c o s α - c o s β} > \frac{α}{β}$ .

Q. If α,β∈(2π​,π) and α<β, then which one of the following is true?

ecosα−cosβ<βα​

ecosβ−cosα<αβ​

ecosα−cosβ<αβ​

ecosβ−cosα<βα​