a) given y=e^-x cos x I found y'=-e^-x(sin x + cos x) and

y"=2e^-x(sin x). I checked and they are right.

b) find stationary points between -pi<x<pi and determine their nature.

I can do this with simple quadratics but this function is beyond me at present.

is always negative ... that means

... now, for what two values of x in the defined interval is that true?
c) Find the points of inflection for -pi < x < pi

same idea ... the only place where y'' = 0 in the given interval is at the endpoints and x = 0 .