devarshi-dt-logo

Question:

Statement I: The equation (sin x)³ + (cos x)³ - aπ³/3 = 0 has a solution for all a ≤ 1/32. Statement II: For any x ∈ R, sin x + cos x = π/2 and 0 ≤ (sin x - π/4)² ≤ 9π²/16. Both statements I and II are true. Both statements I and II are true but I is not an explanation of II. Statement I is true and statement II is false. Statement I is false and statement II is true.

Both statements I and II are true.

Both statements I and II are true but I is not an explanation of II

Statement I is true and statement II is false

Statement I is false and statement II is true.

Solution:

Say f(x) = (sin x)³ + (cos x)³
f'(x) = 0 at x = π/4
f''(x) ≥ 0 at x = π/4 so f(x) = π³/32. This is the least value. Therefore, f(x) ≥ aπ³/3 has a solution. Therefore 1/32 ≥ a. Statement I is incorrect.
Now -π/2 ≤ sin x ≤ π/2
0 ≤ (sin x - π/4)² ≤ 9π²/16
Min at x = π/4 and max at x = π/2. So, Statement I is incorrect and II is correct.