Question

\[ f(x) = \log_e \left( 4x^2 + 11x + 6 \right) + \sin^{-1} \left( 4x + 3 \right) + \cos^{-1} \left( \frac{10x + 6}{3} \right) \]then \( 36|\alpha + \beta| \) is equal to:

Answer 1

Given:
f(x) = ln(4x² + 11x + 6) + sin⁻¹(4x + 3) + cos⁻¹((10x + 6)/3)

Let domain be [α, β]

Step 1: Condition for log
4x² + 11x + 6 > 0
(4x + 3)(x + 2) > 0
⇒ x < −2 or x > −3/4

Step 2: Condition for sin⁻¹
−1 ≤ 4x + 3 ≤ 1
−4 ≤ 4x ≤ −2
⇒ −1 ≤ x ≤ −1/2

Step 3: Condition for cos⁻¹
−1 ≤ (10x + 6)/3 ≤ 1
−3 ≤ 10x + 6 ≤ 3
−9 ≤ 10x ≤ −3
⇒ −9/10 ≤ x ≤ −3/10

Step 4: Intersection of all domains
From step 2 and 3:
x ∈ [−1, −1/2] ∩ [−9/10, −3/10]
= [−9/10, −1/2]

Also satisfies log condition (x < −2 or x > −3/4)
Common region → [−3/4, −1/2]

So,
α = −3/4, β = −1/2

Step 5: Calculate
α + β = −3/4 − 1/2 = −5/4

\[ 36|\alpha + \beta| = 36 \times \frac{5}{4} = 45 \]

Final Answer: 45