What is the Laplace transform of this random variable? Define a random variable that takes only one value for example $$X=c$$ where c is a positive constant.

What does the Laplace of it evaluate to i.e the following $$\mathcal{L}_X(s)= \mathbb{E}[e^{-sX}]= \int e^{-sx} f_X(x)\ dx$$

where $\mathbb{E}$ is the expected value operator.

I have done the following, $$\int ( e^{-sx} ) 1_{x=c} \ dx = e^{-sc}$$

Is my derivation correct?
Yes your analysis is correct. Notice that $X$ is a discrete random variable where $E(e^{X})=\sum_x e^{x}P(X=x)$ where we have $P(X=c)=1$
You can always verify your results by taking the inverse laplace transform of your function.

Here:
$$\mathcal{L}^{-1}(e^{-sc})=\delta(t-c)\text{ where }\delta\text{ is Dirac's Delta Function}$$