For θ ∈[0, π], let f(θ)= sin ( cos θ) and g (θ)= cos ( sin θ) . Let a = max 0 ≤ θ ≤ π f (θ), b = min 0 ≤ θ ≤ π f (θ), c = max 0 ≤ θ ≤ π g (θ) and d = min 0 ≤ θ ∈ π g (θ). The correct inequalities satisfied by a , b , c , d are

Question

For θ ∈[0, π], let f(θ)= sin ( cos θ) and g (θ)= cos ( sin θ) . Let a = max 0 ≤ θ ≤ π f (θ), b = min 0 ≤ θ ≤ π f (θ), c = max 0 ≤ θ ≤ π g (θ) and d = min 0 ≤ θ ∈ π g (θ). The correct inequalities satisfied by a , b , c , d are (A) b< d < c < a (B) d < b < a < c (C) b< d < a < c (D) b < a < d < c

Tardigrade Admin · Accepted Answer

f(θ)= sin ( cos θ) g(θ)= cos ( sin θ) f'(θ)= cos ( cos θ)(- sin θ)<0 ∀ θ ∈[0, π] ∴ f(θ) decreases monotonically ∴ a = max f(θ)=f(0)= sin 1 b = min f(θ)=f(π)=- sin 1 g'(θ)=- sin ( sin θ) cos θ <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/ec856162eac33871d722d9fc688aaa2a-.png" /> g(θ)=1 ; g(π)=1 ; g((π/2))= cos 1 ∴ c= max g(θ)=1 d= min g(θ)= cos 1 ∴ b < d < a < c

Q. For θ∈[0,π], let f(θ)=sin(cosθ) and g(θ)=cos(sinθ). Let a=max0≤θ≤π​f(θ),b=min0≤θ≤π​ f(θ),c=max0≤θ≤π​g(θ) and d=min0≤θ∈π​g(θ). The correct inequalities satisfied by a,b,c,d are

b< d < c < a

d < b < a < c

b< d < a < c

b < a < d < c

f(θ)=sin(cosθ) g(θ)=cos(sinθ) f′(θ)=cos(cosθ)(−sinθ)<0∀θ∈[0,π] ∴f(θ) decreases monotonically ∴a=maxf(θ)=f(0)=sin1 b=minf(θ)=f(π)=−sin1 g′(θ)=−sin(sinθ)cosθ g(θ)=1;g(π)=1;g(2π​)=cos1 ∴c=maxg(θ)=1 d=ming(θ)=cos1 ∴b<d<a<c

Q. For $θ \in [0, π]$ , let $f (θ) = sin (cos θ)$ and $g (θ) = cos (sin θ) .$ Let $a = max_{0 \leq θ \leq π} f (θ), b = min_{0 \leq θ \leq π}$ $f (θ), c = max_{0 \leq θ \leq π} g (θ)$ and $d = min_{0 \leq θ \in π} g (θ)$ . The correct inequalities satisfied by $a, b, c, d$ are