Given , then for all real values of
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: B
Solution
A = sin²θ + cos⁴θ = sin²θ + (1 - sin²θ)². Let t = sin²θ, so t ∈ [0,1].
A(t) = t + (1-t)² = t + 1 - 2t + t² = t² - t + 1.
This is a upward parabola with minimum at t = 1/2: A = 1/4 - 1/2 + 1 = 3/4.
At endpoints t = 0 or t = 1: A = 1.
So A ∈ [3/4, 1], which is option 2.
A(t) = t + (1-t)² = t + 1 - 2t + t² = t² - t + 1.
This is a upward parabola with minimum at t = 1/2: A = 1/4 - 1/2 + 1 = 3/4.
At endpoints t = 0 or t = 1: A = 1.
So A ∈ [3/4, 1], which is option 2.
[1980]