At least in Delaware law, a board's fiduciary duty involves a duty of care, loyalty, and good faith. While acting to maximize their personal interests or egregiously working against profits could be considered a breach, not trying to maximally make profits probably wouldn't be.
All of this would need to be litigated on a case-by-case basis, but many modern cases have found that companies can freely act to maximize the wages of their employees at the expense of dividends or act charitably even when there aren't tax breaks.
That said, I'm not sure this really even applies in the case of Twitter---I think its not unreasonable to argue that the innate value of Twitter is so much higher than what Musk is offering that its better for the shareholders to wait even thinking purely in terms of profits (I would probably personally disagree with that, but I don't think its any more unreasonable than plenty of other valuations I see)
Yes, they don't have to maximize profits, but they also can't ignore profit to the benefit of other values they deem more important, per the very old but still in force Dodge v Ford.
As long as they are not egregiously and obviously acting against that goal, or taking decisions without deliberation, the law would favor them in any trial that only hinged on whether the price offered by Musk was better than the market. Basically the burden of proof to show that it could not have been a good business decision would lie with the plaintiff, and I very much doubt that you could build a solid case of that.
I mean, saying "We are concerned about a hostile takeover attempt by a person who isn't allowed to be an officer of a public company anymore because he has show himself either unwilling or unable to be a fiduciary for stockholders, and we are trying to protect our stockholders from him" seems like a slam dunk in court.
Musk is offering to buy out all stockholders, so if the sale goes through, he will be the sole owner of Twitter, and all current stock holders will receive cash.
On the other hand, the board can easily say "we don't beleive this deal offered by a person who isn't allowed to be a public officer of a company anymore will ever go through, either because he will retract it or because he will be unable to secure funding; and we beleive attempting the deal will hurt stock prices and profits for current shareholders" and then I agree, it would probably be a slam dunk in court.
I don't think Dodge v. Ford isn't a great case to cite here for two reasons:
1) I think the trend of accepting that corporations can exist for reasons other than to generate profit is a post-ww2 phenomenon and this case is from 1919
2) Perhaps more importantly, Dodge v. Ford was a case before the Michigan Supreme Court and, to my knowledge, has never been used in a case in Delaware.
All of this would need to be litigated on a case-by-case basis, but many modern cases have found that companies can freely act to maximize the wages of their employees at the expense of dividends or act charitably even when there aren't tax breaks.
That said, I'm not sure this really even applies in the case of Twitter---I think its not unreasonable to argue that the innate value of Twitter is so much higher than what Musk is offering that its better for the shareholders to wait even thinking purely in terms of profits (I would probably personally disagree with that, but I don't think its any more unreasonable than plenty of other valuations I see)