Vedanta Sutra 1.1.1

ब्रह्म जिज्ञासाधिकरणम् 

अथातो ब्रह्मजिज्ञासा (i.1.1)

अथ - Now/Thereafter  after having acquired eligibility (अत्र अपि आनन्तर्यं एव सक्षदर्थः अधिकारस्तु लभ्यते (Srila Vadiraja Tirtha) . अथ also denotes auspiciousness like sound of conch
अतः - Henceforth (shows purposiveness of जिज्ञासा ) हेतु भावयोर् भवतः (Srila Baladeva Vidyabhushana)
ब्रह्म जिज्ञासा - inquiry into Brahman

Thereafter henceforth inquiry into the characteristic features of the supreme/Brahman. 

The context and line of reasoning:

अपाम सोममम् अमृता अभूम अक्षयं ह वै चातुर्मास्य यजजिनः सुकृतम् भवतीति आदिशु धर्मैर अमृतत्वाक्षय्य सुखत्वा श्रवणान् न युक्तेति पूर्वास्मिन् पक्षे प्राप्ते भगवान् बादरायणो व्यासः प्रारिप्सितसय् शास्त्रस्यादिमम् सूत्रम् इदं अवतारयति
To the objection, that inquiry into Brahman is not necessary as it is said in the Vedas that we have obtained immortality by drinking soma juice and those who follow chaturmasya vrata have obtained sukruti, in anticipation of this objection Bhagavan Badarayana Vyasa manifests the first sutra

Some points from commentaries by Acharyas: 

Srila Sankaracharya(S): one who has the i) the right discrimination between eternal and non-eternal ii) right dispassion -non attachment to enjoyment of fruits iii) right conduct - equanimity, abundance of calmness  iv) right earnestness to know Brahman or desire for release is eligible for studying the Vedanta sutras. These 4 qualities are known as  साधन चतुर्लक्षण्यम्

Srila Ramanujacharya (R): one who has understood that the results of karma kanda is temporary, limited, etc. becomes eligible for enquiry to Brahman

Srila Madhvacharya (M): gives the understanding by taking it's four fold requisites:

• अधिकारी: one based on a) intrinsic classification of souls (Devas, Rishis and men) and other based on b) earned merit. Based on earned merit qualifications necessary are Vedic study, tranquility, self control, contentment, concentrations etc and realization of the impermanence of everything except God and  परत्रन्त्रत्व of everything on God etc.
• विषय- The supreme Being of infinite perfections
• प्रयोजन -  moksha 
• संबन्ध - By His grace the jiva is delivered 

In his Brahma Sutra Bhasya, sums it up well in his Brahma Sutra Bhasya as
यतो नरायणप्रसादमृते  मोक्षः न च ज्ञानं विना अत्यर्थप्रसदः अतो ब्रह्मजिज्ञासा कर्तव्य ||
No moksha without the grace of Narayana, no fullness of grace without understanding of Him, so inquiry into Brahman/Narayana is our duty

Srila Baladeva Vidyabhushana (BV) :
न चात्र कर्म संपत्ति आनन्तर्यम् शक्यं वक्तुं , तद्वताम् अपि सत्संग विरहिणाम् ब्रह्म जिज्ञासया अदर्शनात् | तत् शून्यं अपि सत्याद्-पूतानाम् सत् प्रसङ्गिनाम्  दर्शनात् च
It is not that only after having acquired karma (studied/mastered karma kanda) that one becomes eligible, because it is seen that those who have mastered karma kanda but who are bereft of holy association do not inquire into Brahman. But it is seen amongst those who have not studied Karma Kanda but have been purified by pious acts in association with holy people (devotees)

न च नित्यानित्य विवेकदी साधन चतुष्तय सम्पत्त्यानन्तर्यम् शकयं वक्तुं  | प्राक् तस्य दौर्लभ्यात् सत् प्रसंग  शिक्ष पर भाव्यत्वात्त च
It is not that one after having acquired the 4 qualities becomes eligible, but it is almost impossible to attain the qualities, but can be obtained by getting the teachings of the holy devotees.

तस्मात् तादृशस्य पुंसस तदनन्तरम् तत् जिज्ञासा युक्त इत
Thus after having attained the four qualities by the association of the devotees he is endowed with this spirit of inquiry.

Verses from Srimad Bhagavatam which correspond with this Sutra.


Srila Bhakti Srirupa Siddhanti Maharaj: 11.18.38-39, 11.3.21

Srila Hari das Shastri Maharaj: 1.2.8-11, 1.1.1.

Personal Notes: 

SB 11.3.21
दु:खोदर्केषु कामेषु जातनिर्वेद आत्मवान् ।
अजिज्ञासितमद्धर्मो मुनिं गुरुमुपव्रजेत् ॥ ३८ ॥

SB 11.3.21
तस्माद् गुरुं प्रपद्येत जिज्ञासु: श्रेय उत्तमम् ।
शाब्दे परे च निष्णातं ब्रह्मण्युपशमाश्रयम् ॥ २१ ॥

Srila Madhvacharya points out the contradiction in S's commentary. If person has the 4 qualities, i.e., he can differentiate between nitya and anitya then according to Advaita he should already have attained the  highest state of having realized he is Brahman, so then where is the question and necessity of Brahma jijnasa. 

References:

1. Vedanta Sutra by Srila Bhakti Prajnana Yati Maharaj
2. Govinda Bhasya sukhsma teeka - Srila Bhakti Srirupa Siddhanti Maharaj
3. Brahma Sutras and their principle commentaries - BNK Sharma
4. Brahma Sutras - HH Bhanu Swami
5. The Vedanta sutras of Badarayana - Srisa Chandra Basu