Exxon Mobil’s 25-year “Outlook for Energy,” an internal planning document, takes the trends of recent years and projects them forward, modeling the company’s worldview in a neat line that shows global energy consumption rising 1.2 percent a year for decades to come. By 2030, the world will use almost 40 percent more energy than it did in 2005, Exxon Mobil predicts. It says fossil fuels—oil, natural gas, and coal—will continue to meet about 80 percent of demand and will emit 28 percent more carbon dioxide into the atmosphere than they do today. In Exxon Mobil’s forecast, renewables—wind, solar, and biofuels—will supply just 2 percent of the world’s energy needs by 2030. With the company seeing such a minuscule market for renewables, Tillerson has publicly proclaimed, “We haven’t found an alternative to invest in.”

Critics wonder if that comment will one day occupy a special place in corporate infamy, alongside other legendary scoffs. In the mid-19th century, Telegraph Co., the ancestor of Western Union, monopolized the national telegraph system. But when Alexander Graham Bell and his business partner offered to sell Bell’s telephone patents to Telegraph, executives there couldn’t be bothered by the “idiotic” technology. A Telegraph Co. committee reported its findings about Bell’s telephone proposal to company president Chauncey DePew: “Technically, we do not see that this device will be ever capable of sending recognizable speech over a distance of several miles.” And then there was Ken Olsen, founder and longtime president of mainframe-computing pioneer Digital Equipment Corp., who, in 1977, famously sniffed, “There is no reason for any individual to have a computer in his home.”

“Look at Ford,” says Peter Schwartz, an author and business consultant who headed scenario planning for Shell in the 1980s. Just a decade ago, Ford Motor had an environmentally friendly CEO in William Ford Jr. Yet without regulatory pressure to produce more fuel-efficient cars, the company “missed the window to adapt,” Schwartz says. “The question now for Exxon Mobil, given its scale, is will it adapt in time?”

Some Exxon Mobil shareholders worry that time is running out—an astounding notion given the company’s outsize profits. Members of the Rockefeller family—descendants of founder John D. Rockefeller—say they think that the Tillerson way is a case of corporate tunnel vision that is as reckless as it is skewed.

John D. Rockefeller’s great-granddaughter Neva Rockefeller Goodwin, a development economist at Tufts University, confronted Tillerson and the board at the company’s most recent annual meeting, in Dallas. Goodwin, 64, described what she sees as a potentially cataclysmic flaw in two of Exxon Mobil’s fundamental business assumptions. The company’s management, she said, assumes that developing economies will continue to grow rapidly and that this growth will drive increasing demand for fossil-fuel energy sources, ignoring the fact that such changes will cause climate shifts that will hit developing countries particularly hard. “These assumptions cancel one another out,” Goodwin argued, as Tillerson and his board listened patiently.

Goodwin’s resolution, which called on the company to investigate these “internal contradictions” and to take a leadership role in developing sustainable energy, garnered votes from approximately 10 percent of those Exxon Mobil shareholders who voted. Another Goodwin-supported proposal, to replace Tillerson with an independent, nonexecutive chairman, garnered 40 percent. Undeterred, the company’s opponents are still circling.

In his campaign for the presidency, Obama mentioned Big Oil in general and Exxon Mobil in particular dozens of times, and not a single reference was friendly. His choice for energy secretary, Steven Chu, is a Nobel Prize winner on the vanguard of alternative-energy research.

During the election campaign, Obama openly mocked Tillerson’s company. “It’s a game where lobbyists write check after check, and Exxon turns record profits, while you pay the price at the pump and our planet is put at risk,” the new president railed during the primaries. He also inveighed against windfall oil profits, singling out Exxon Mobil several times as a corporate predator. “They are not going to give up those profits easily,” said candidate Obama.

In addition to Chu, Obama has packed his administration with other hawks on climate change, including Harvard University physicist John Holdren, the new White House science adviser, who has lambasted Exxon Mobil for funding climate-change skeptics.

David Sandalow of the Brookings Institution, formerly part of Obama’s transition team for energy policy, published a book last year called Freedom From Oil: How the Next President Can End the United States’ Oil Addiction, in which he argued for a massive federal push to put millions of electric-powered cars on U.S. roads. Obama took up that cause in February’s stimulus bill, which offers tax breaks for buyers of flexible-fueled and hybrid-electric cars and commits billions of dollars for heavy-duty-battery research and development. “The future of the auto industry is electrification,” says Sandalow. “The only question is how quickly.”

Every investment at Exxon Mobil, whether it’s $100,000 or $100 million, must pass a rigid review. This rigor gives Exxon Mobil its edge over rivals, one company executive says. Tillerson’s predecessor, the legendary Lee Raymond, who was chairman and CEO from 1993 through 2005, would openly mock other oil companies for writing off billions of dollars of shareholder capital from bad investments.