Let f:[0,2] arrow R be the function defined by f(x)=(3- sin (2 π x)) sin (π x-(π/4))- sin (3 π x+(π/4)) If α, β ∈[0,2] are such that x ∈[0,2]: f(x) ≥ 0 =[α, β], then the value of β-α is .

Question

Let f:[0,2] arrow R be the function defined by f(x)=(3- sin (2 π x)) sin (π x-(π/4))- sin (3 π x+(π/4)) If α, β ∈[0,2] are such that x ∈[0,2]: f(x) ≥ 0 =[α, β], then the value of β-α is . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let π x=θ f(x)=(3- sin 2 θ) sin (θ-(π/4))- sin (3 θ+(π/4)) =(3- sin 2 θ) (( sin θ- cos θ)/√2)-(( sin 3 θ/√2)+( cos 3 θ/√2)) =(1/√2)[(3- sin 2 θ)( sin θ- cos θ)-(3 sin θ-3 cos θ-4 sin 3 θ+4 cos 3 θ)] =(1/√2)[(3- sin 2 θ)( sin θ- cos θ)- 3( sin θ- cos θ)-4( sin θ- cos θ) ..(1+( sin 2 θ/2)) ] =(( sin θ- cos θ)/√2)(3- sin 2 θ+1+2 sin 2 θ) ∴ For f(x) ≥ 0 <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/bd4bd52d550af2c9949fd882b23865f8-.png" /> ∴ [α, β]=[(1/4), (5/4)] ⇒ [β-α]=(5/4)-(1/4)=1

Q. Let f:[0,2]→R be the function defined by f(x)=(3−sin(2πx))sin(πx−4π​)−sin(3πx+4π​) If α,β∈[0,2] are such that {x∈[0,2]:f(x)≥0}=[α,β], then the value of β−α is ______.

Q. Let $f : [0, 2] \to R$ be the function defined by $f (x) = (3 - sin (2 π x)) sin (π x - \frac{π}{4}) - sin (3 π x + \frac{π}{4})$
If $α, β \in [0, 2]$ are such that ${x \in [0, 2] : f (x) \geq 0} = [α, β]$ , then the value of $β - α$ is ______.