Hello, could you help solve this problem step by step with well-formatted formulas? Hand written answers are also welcome!

If X1,...,Xn is a random sample with size n>2 from a exponential distribution with the p.d.f below:

f(x)={1ex0x0,x<0

Let >0 is unknown,

(a) How should we show that the distribution of the data has monotone likelihood ratio in T=X1+...+Xn ?

(b) If 0>0 is some specific value of interest, what is the UMP test ofH0:0 versusH1:>0 at some level of significance 0.05? (Xi has a gamma distribution )