Let m and M respectively be the minimum and the maximum values of f(x) = sin ⁻¹2x + sin 2x + cos ⁻¹2x + cos 2x, x ∈ [ 0,π/8 ]. Then m + M is equal to :

Solution

Maths sir · 4 steps

Pair the inverse terms first: sin⁻¹ 2x + cos⁻¹ 2x is just π/2, a constant.

1. Inverse terms combine. For x ∈ [0, π/8], 2x lies between 0 and about 0.785, inside [-1, 1]. So sin⁻¹ 2x + cos⁻¹ 2x = π/2, and f(x) = π/2 + sin 2x + cos 2x.

   sin 2x + cos 2x = √2 sin(2x + π/4)

2. Track the angle. As x goes from 0 to π/8, the angle 2x + π/4 goes from π/4 to π/2. Sine is increasing there, so sin 2x + cos 2x rises from √2 × 1/√2 = 1 to √2 × 1 = √2.
3. Minimum and maximum. m = f(0) = π/2 + 1 and M = f(π/8) = π/2 + √2.
4. So. m + M = π + 1 + √2, which is option (a).

The idea to keep. Whenever sin⁻¹ t and cos⁻¹ t appear with the same argument, replace the pair by π/2. Then write a sin θ + b cos θ as one sine and check its range on the given interval.

Answer from the NTA final answer key. Solution by PrepWiser.