The greatest value of (2 sin θ+3 cos θ+4)3 ⋅(6-2 sin θ-3 cos θ)2, as θ ∈ R, is

Question

The greatest value of (2 sin θ+3 cos θ+4)3 ⋅(6-2 sin θ-3 cos θ)2, as θ ∈ R, is (A) 2345 (B) 3456 (C) 1234 (D) 4567

Tardigrade Admin · Accepted Answer

Clearly, 2 sin θ+3 cos θ+4 > 0 ∀ θ ∈ R and 6-2 sin θ-3 cos θ > 0 ∀ x ∈ R Put. 2 sin θ+3 cos θ+4=x and 6-2 sin θ-3 cos θ=y Now, x+y=10 Also, A.M ≥ G.M. (for positive numbers) ∴ ((x/3)+(x/3)+(x/3)+(y/2)+(y/2)/5) ≥((x3/27) ⋅ (y2/4))(1/5) ∴ (2 sin θ+3 cos θ+4)3(6-2 sin θ-3 cos θ)2 ≤(32)(27)(4)=3456

Q. The greatest value of (2sinθ+3cosθ+4)3⋅(6−2sinθ−3cosθ)2, as θ∈R, is

2345

3456

1234

4567

Clearly, 2sinθ+3cosθ+4>0∀θ∈R and 6−2sinθ−3cosθ>0∀x∈R Put. 2sinθ+3cosθ+4=x and 6−2sinθ−3cosθ=y Now, x+y=10 Also, A.M≥G.M. (for positive numbers) ∴53x​+3x​+3x​+2y​+2y​​≥(27x3​⋅4y2​)51​ ∴(2sinθ+3cosθ+4)3(6−2sinθ−3cosθ)2 ≤(32)(27)(4)=3456

Q. The greatest value of $(2 sin θ + 3 cos θ + 4)^{3} \cdot (6 - 2 sin θ - 3 cos θ)^{2}$ , as $θ \in R$ , is