Three of the four shareholder resolutions backed by the Rockefellers concern global climate change. The fourth seeks to have the chairman's role be independent from that of the chief executive. Rex Tillerson became C.E.O. and chairman of Exxon Mobil in January 2006. (Exxon Mobil recommends that shareholders reject all four proposals.)

Rockefeller family members had quietly brought up these issues with company executives in years past, said Goodwin's cousin Peter O'Neill, a great-great-grandson of John D. "It was not an easy decision for the majority of the Rockefeller family to go public with our concerns," he said.

Goodwin hailed Exxon Mobil's "strong corporate culture," noting that its top executives have spent their entire careers at the company. But, she added, that culture "includes not listening to outside voices."

Goodwin recalled an informal lunch that she had with Lee Raymond, then chairman and chief executive of Exxon Mobil, and Tillerson who was about to succeed him. "I was told I had to behave myself and not say much," she said.

There are 152 descendants of John D., who died in 1937. Seventy-eight of them are over the age of 21, and 66 are supporting the four shareholder resolutions, O'Neill said.

The Rockefeller cousins, he said, do both business and philanthropy together and "spend a lot of time" communicating with one another about common interests.

The Rockefellers could not say how much of Exxon Mobil is owned by family members. It has been estimated that the combined family stake is not very large, but Goodwin said that it "is a significant holding for us."

When the U.S. Supreme Court ordered the breakup of Standard Oil as an "unreasonable monopoly" in 1911, it was split into 34 companies. One, Standard Oil of New Jersey, became Exxon; another, Standard Oil of New York, became Mobil. The Rockefeller family had shares in all 34 companies as a result. Exxon and Mobil merged in an $80 billion deal in 1998.

The news conference came a day before Exxon Mobil is expected to report a blowout first quarter on higher oil prices. Indeed, the Rockefellers were careful to emphasize that they were pleased with Tillerson and the recent performance of the company and the stock price.

"When Exxon Mobil does well, so do I and other members of my family," Goodwin says. But she and O'Neill said that family members are concerned that the company is not doing enough to prepare for the long term, as some of its rivals have.