In Mansfield Park how and how far does Jane Austen challenge her readers to examine values?
All the various definitions of "challenge" include the concept of intentional antagonism of one sort or another, for example, challenge to take up a dispute, to prove/disprove the truth or validity of something, to win at a race or struggle, etc. There is always an element of antagonistic intent to show someone or something wrong, inadequate, invalid, etc. While Bronte nay be said to be striking out in a challenge against society's values in Jane Eyre or Harper Lee in To Kill a Mockingbird, because of their bitter and troubled narratorial tones, the pleasant, ironic, amusing, lighthearted and sympathetic narratorial tone in Mansfield Park seems to belie any suggestion of "challenging" readers in any way to do anything. In other words, the gentle though ironically witty narrator of Mansfield Park, recognized as Jane Austen's own voice, shows by her tone that while she may be laying open the inner foibles and transgressions of upper class society, she is doing it with the loving touch of a fond niece telling tales about an eccentric though beloved aunt: there is no antagonism thus no challenge in the narrator's tone or expression.
Having said this, it is clear that Austen does lay the social values of her part of the upper class world open to view and that part of what she makes visible on her "two inches of ivory" is the sets of values of various sets of characters and of individual characters. The way she does this, the "how," is through psychological development of characters. She reveals their modes of thinking and their rationales and motives for actions and choices. An example of this comes early in the novel in the conversation about what to do for the outcast sister and her multitude of children. Each of the three principle characters expresses their thoughts and reasonings on the subject thus exposing their values both directly and indirectly through their direct statements or through the implications of their statements. An illustration of this is Mrs. Norris's double standard relating to caring for the eldest daughter they all agree to take charge of. Mrs. Norris advances the scheme and speaks of the moderate expense then rejects all notion of personally participating in raising her:
in reply to Lady Bertram's calm inquiry of "Where shall the child come to first, sister, to you or to us?" Sir Thomas heard with some surprise that it would be totally out of Mrs. Norris's power to take any share in the personal charge of her. ... Mrs. Norris was sorry to say that the little girl's staying with them, at least as things then were, was quite out of the question.
As to the question of "how far" a challenge from Austen might extend in provoking her readers to examine their social values, if there is no challenged proffered, then it cannot be said that the challenge produces a deep or a shallow effect. What Austen does do is to open the psychological motivation and reasoning of several different types of upper class men and women, thus painting a very detailed and vivid picture of society's values for readers to reflect upon.