Question

The set of all values of k for which (tan^–1 x)³ + (cot^–1 x)³ = kπ³, x ∈ R, is the internal ______.

पर्याय

• `[1/32, 7/8)`

• `(1/24, 13/16)`

• `[1/48, 13/16]`

• `[1/32, 9/8)`

Answer 1

The set of all values of k for which (tan^–1 x)³ + (cot^–1 x)³ = kπ³, x ∈ R, is the internal `underlinebb([1/32, 7/8)`.

Explanation:

Given expression is (tan^–1 x)³ + (cot^–1 x)³.

= (tan^–1 x + cot^–1 x)

((tan^–1 x)² + (cot^–1 x)² – tan^–1 x cot^–1 x)

= `(π/2)((tan^-1x + cot^-1x)^2 - 3tan^-1x cot^-1x)`

= `π^3/8 - (3π)/2 tan^-1 x(π/2 - tan^-1x)`

= `(3π)/2(tan^-1x - π/4)^2 + π^3/32`

Above expression will have a minimum value when `(tan^-1x - π/4)` becomes 0 at x = `π/4`.

So, the minimum value is `π^3/32`.

`\implies tan^-1 x ∈((-π)/2, π/2)`

Therefore, the maximum value occurs at x = `-π/2`.

So, the maximum value is `(7π^3)/8`.

= `π^3/32 ≤ kπ^3 < 7/8π^3`

= `1/32 ≤ k < 7/8`

Required range = `[1/32, 7/8)`