Is this the world's sexiest woman (and the most powerful)?

In May 1968, when Christine Lagarde was a teenager, French schools were shut down during a student uprising. As her fellow pupils took to the streets to throw cobblestones at the police, Lagarde took up synchronised swimming and went on to win a bronze medal in the national championships. "It was synchronised swimming that taught me: 'Grit your teeth and smile'," she has said. And it's been helpful in her subsequent career in politics: "In exactly the same way, it's a sport of resistance and endurance. You're in tension and control."

If her teeth are gritted, it's impossible to tell. What lovely teeth she has – straight and white, they gleam out of a permanently, almost alarmingly, tanned face. Tall – she's 5ft 10in – and slim, the 55-year-old Lagarde dresses with the casual élan of a Parisian, patriotically attired in Chanel suits and Hermès scarves, along with jazzy bracelets and fur-lined ponchos. Lagarde softens her rather severe black-and-white outfits with silk scarves, a string of pearls or a brooch. She has widely spaced green eyes framed by a silver bob. She still swims, but not in formation.

Two weeks ago Lagarde became the first woman to head up the International Monetary Fund (IMF), the latest high in a career that has seen her become the first female finance minister of a G8 economy, as well as the first female chair of an international law firm. In a country where a minister has recently said that politics is so sexist she can't wear a skirt in parliament without attracting comments, Lagarde's rise is extraordinary, and it's partly because she's stayed away from France. She is keen on helping women, but she's also determinedly different from many women in French politics, who sometimes have a way of undermining themselves.

"She's unusual among French female politicians in that there's nothing coquettish about her," says Andrew Hussey, a professor at the University of London Institute in Paris. "A lot of the others – such as Ségolène Royal – play on a kind of French feminine elegance."

Lagarde, by contrast, creates an almost surreal aura of veneration. This is combined with a nuanced command of English, a precise intellect and unnerving stamina. People talk about Lagarde with admiration that borders on hero worship. She "radiates charm", she "oozes respect"; she's described variously as intelligent, beautiful, upstanding and elegant. In short, they usually conclude, elle a de la classe.

In her campaign to get her new job, Lagarde had to convince China that she was worth voting for. The emerging market economies had spent weeks complaining about the European stranglehold on the job, but Lagarde won round Beijing in an afternoon. Lunch with the central bank governor and a meeting with the deputy prime minister was enough to clinch it.

Lagarde was born on New Year's Day 1956 and grew up in Le Havre in northern France. She had three younger brothers; her father was an English professor, her mother a Latin teacher. They hosted intellectual dinner parties and took her to the opera. Her father died of motor neuron disease when she was 17, and soon afterwards she went to study in America, before returning to France to study law. After twice failing the exams to get into the French civil service and being told she would never make partner in a French law firm as a woman, she joined the international firm Baker & McKenzie, where she rose to chairman of the board. She used the same title she's taking at the IMF: Madame Chairman. Syntax, she explained at the time, is not the battleground on which the feminist cause will be gained or lost: "I didn't want to find a feminine equivalent to chairman. Insisting on marking femininity by the gender of words is ridiculous."

When Dominique de Villepin, the then French prime minister, asked her to come back to France to join the government in 2005, she left Chicago in such a hurry that she left her glasses behind. She has said she had no idea who was sitting around the table at the next day's meeting. After two days in government, she committed one of her few political blunders. Having spent several years in Chicago, she was astonished by how much French economic debate focused on holidays and days off, and she said that France's venerated labour laws should be changed. The press promptly dubbed her Madame La Gaffe. The nickname didn't stick, and by the end of her time in the finance ministry she'd even persuaded the French to take a more flexible approach to their 35-hour working week.

Lagarde's charm and her international contacts book became very valuable to the French government. She pleased her country by persuading the German government to drop its objections to a reduced VAT in French restaurants and she led calls to impose taxes on bankers' bonuses. On the night when the American insurer AIG could have collapsed, she rang up Henry Paulson, the then US treasury secretary. "I had just 15 seconds to persuade him not to let it collapse," she has said.

It's not just the powerful whom she charms. When Lagarde visited Chicago earlier this year, she went to visit the apartment building where she lived six years ago and the janitor came running out and kissed her and told her how much they all missed her. "She has an American informality, which she combines with a sense of distance," says Dominique Moisi, a founder of the French Institute for International Relations and an acquaintance. "She knows how important she is, but she's quite warm in spite of everything. She can be funny when she knows she needs to be funny."

Still, that combination of power and self-control can seem a little sinister. "If I were to use two words to describe her they would be independence and self-control," says Moisi. Grit your teeth and smile. Lagarde is a teetotal vegetarian; she goes to the gym every day, cycles 20 to 30km once a week and swims as often as she can. She told a French paper recently that "success is never complete. It's an endless combat. Each morning one must put one's capacities to the test once again."

Law, French politics and international finance are still some of the professions most weighted against women: fewer than 20% of French MPs are female. "For a woman at this level it's probably more difficult than for a man," says Bruno Silvestre, Lagarde's former spokesman at the finance ministry. "Christine has to be as good as a man, which probably means better than a man." In meetings, she listens to people very hard. Lagarde has described listening as a female virtue, and often warned against "hairy-chested showing off" in international politics. At Baker & McKenzie, Eduardo Leite, who was on the board with her, says that Lagarde used to say to female lawyers: "Don't try to imitate the boys. Be yourselves and support each other." She doesn't flirt, but she does use her femininity, and she's on kissing terms with many more people than the average French minister.

Lagarde has been married and divorced twice. With her first husband, Wilfried, she had two sons, now in their mid-20s; while living in Chicago she fell in love with a British businessman, Eachran Gilmour, and married him. Then in 2006, when she was exterior commerce minister, she went on a trip to Marseille to meet local entrepreneurs. One of them was her old friend from law school, a Corsican businessman named Xavier Giocanti and, they say, it was a coup de foudre. Lagarde has been with him ever since. "She radiates," says Moisi. "It's even the way she walks – I think that's because she swims so much. She's more special today than she was yesterday. She's beautified with age. She's stunning – she's not Angelina Jolie, but she's stunning."

It can all get a little odd. A French politician talked fondly to me of Lagarde's overdeveloped, swimmer's shoulders, but most people stop short of describing her in overtly sexual terms. Lagarde has often said it's important not to have too much testosterone in one room, but in terms of the personal impression she makes, she seems to have achieved the feminist ideal: the men in the boardroom admire her looks chastely. She is une grande femme, a distinguished woman with an air of authority – and nice shoulders.

Since Lagarde is a French politician, there is of course an affaire attached to her name – the "Tapie affair" – and French judges are deliberating on whether to open an investigation into claims that she abused her role as finance minister in 2007. Businessman Bernard Tapie was claiming damages from the bank Crédit Lyonnais, which he claimed had mishandled his sale of the sportswear firm Adidas, but the bank had already been liquidated and a state-owned consortium was in charge of its liabilities. Lagarde stepped in to end a 14-year court dispute in 2007 by ordering a panel of judges to arbitrate the case, and the following year Tapie, a friend of Sarkozy, was awarded ¤285m in damages, payable by the French state.

Lagarde's opponents say she interfered in the justice system on behalf of Tapie. Silvestre points out that the government risked facing an even bigger bill if things had carried on any longer. Even the MP Charles de Courson, who ruled, as a member of the panel set up to oversee the consortium, that the judges' decision was a serious mistake, says that Lagarde herself is not to blame. "She acted on orders," he says. "She's done nothing wrong in her life." She will find out if there is to be an investigation early next month.

Lagarde has said she does not regret her decision. "She's been a faithful servant of the president [Sarkozy], but she's never behaved like a courtesan," says Moisi – and that's the kind of thing people say in French politics, as if that's a remarkable achievement. Lagarde has taken more of an Anglo-Saxon approach to a male-dominated profession. She likes to quote Madeleine Albright, the first woman to become a US secretary of state – "There's a special place in hell for women who don't help other women" – and Eleanor Roosevelt: "Women are like tea bags. When you put them in hot water you find out how strong they are."

She's keen on the feminine virtues, then, without being po-faced. At the finance ministry she used to keep a collection of cartoons. Her favourite shows her in fishnet stockings, whipping a banker.