It is estimated that 1kg of body fat will provide 3.8 * 10^7 j of energy. a 67kg mountain climber decides to climb a mountain 3500 m high how much work is done by gravity on the climber for the trip up to the top of the mountain

During a climb UP the mountain, gravity does NO work on the climber.
Actually, it's more correct to say that gravity does NEGATIVE work
on him.  The climber has to DO the positive work to haul himself up.
  
                     Work = (mass) x (gravity) x (height) .

For the guy in this problem:

                     Work = (67 kg) x (9.8 m/s²) x (3,500 meters)

                             =  2,298,100 joules.

If he eats no candy bars on the way, and completely depends on
his stored body fat for the energy, then he'll burn off

                       (2,298,100 joules) / (3.8 x 10⁷ joules/kg)

                   =          0.06 kg of fat.

That's only about 2.1 ounces.  We KNOW he'll lose more weight than that,
climbing 11,000 feet.  That's because climbing is pretty inefficient. 
In addition to the potential energy you have to give your body weight,
you also have to expend energy breathing, digesting, metabolizing,
and sweating.