Question

The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.

Options

• `[-4/3, 2]`

• [1, ∞)

• (–∞, –1]

• (–3, –1)

Answer 1

The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is `underlinebb([-4/3, 2]`.

Explanation:

Given that f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2); x ≠ 2nπ, n ∈ N`

f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)(cos  x/2)(sin  x/2)`

f(x) = (4a – 3)(x + log_e5) + (a – 7)(sinx)

Also given that f(x) has critical points,

i.e., f'(x) = 0

(4a – 3)(1 + 0) + (a – 7)(cosx) = 0

cosx = `(3 - 4a)/(a - 7)`

As, `cosx∈[–1, 1]`

`(3 - 4a)/(a - 7) ∈[-1, 1]`

`-1 ≤ (3 - 4a)/(a - 7) ≤ 1`

`(3 - 4a)/(a - 7) ≥ -1` and `(3 - 4a)/(a - 7) ≤ 1`

`(3 - 4a)/(a - 7) + 1 ≥ 0` and `(3 - 4a)/(a - 7) - 1 ≤ 0`

`(3 - 4a + a - 7)/(a - 7) ≥ 0` and `(3 - 4a - a + 7)/(a - 7) ≤ 0`

`(-3a ≥ 4)/(a - 7) ≥ 0` and `(10 - 5a)/(a - 7) ≤ 0`

`(a + 4/3)/(a - 7) ≤ 0` and `(a - 2)/(a - 7) ≥ 0`

`a∈[(-4)/3, 7)` and `a∈(-∞, 2] ∪ (7, ∞)`

`a∈[(-4)/3, 2]`