Fun with Legendre transforms: a. Show the validity of the Lagendre transform: Given a function f(x), define y = and g = f-yx with g being regarded as a function of y, show that=-x. b. Take f(x) = where x is restricted to be nonegative. i. Find explicit expressions for y(x) and its inverse expression for x(y). ii. Find g(y) iii. Verify explicit using i. and ii that= -x