ikelihood

#1
hello, I have a question about the topic ikelihood.
In the literature, the likelihood function is used for the estimation of the parameters, and in practice the definition is the following:
It's called likelihood function associated with the sample result X = x0 the function L: OMEGA -> [0, INF) which associates to each value of the theta parameter of a probability density function f (x. | Theta).
but I saw some exercises where the likelihood is used for troubleshooting. example
"Let x be a sample of numerosit`a bulbs n = 14, the lifetime of which is supposed exponential parameter?.
the timing values ​​<- c were observed (0.14,1.46,49.99,17.30,11.88,70.96,20.87,4.90,9.55,9.19,11.30, 48.42,36.99,4.60);
determine the maximum likelihood estimator. (I ask myself: for what purpose?);
also provide an estimator for the average duration of life of the bulbs.
My question is, in this context I could have used other methods of estimation? as for example bootstrap? because it uses the likelihood?
you can use the likelihood function in official publications? I have never seen.