I was trying to solve the following question from Sheldon Ross, Introduction to Probability Models.
The number of missing items in a certain location, call it $X$, is a Poisson random variable with mean $\lambda$. When searching the location, each item will independently be found after an exponentially distributed time with rate $\mu$. A reward of $R$ is received for each item found, and a searching cost of $C$ per unit of search time is incurred. Suppose that you search for a fixed time $t$ and then stop. i) Find your total expected return. ii) Find the value of $t$ that maximizes the total expected return.
In order to solve i), I have done the following:
Let $Z$ be an exponential variable with parameter $\mu$. Define the collection of independent identically distributed indicator random variables $$Y_{i}=\left\{\begin{array}{ll} 1 & \text { if }Z<t \\ 0 & \text { if }Z>t \end{array}\right. $$ $Y_i$ essentially denotes if the $i^{th}$ item has been found. Then, if $X(t)$ denotes the total number of items found in time $t$, it follows that $$X(t):=\sum_{i=1}^{N(t)}Y_i$$ where each $Y_i$ is iid and $N(t)$ is a Poisson process with parameter $\lambda$. Then, $$E[X(t)]=\lambda t E[Y_1]=\lambda t P(Z<t)=\lambda t(1-e^{-\mu t})$$ and so $$E[RX(t)-Ct]=RE[X(t)]-Ct=R\lambda t (1-e^{-\mu t}) -Ct.$$
My problem is the fact that ii) requires me to maximise the value of $t$ obtained above. When I plot the graph or put it on Wolframalpha, it appears that there is no global maxima. So I am not sure if the expectation I calculated is wrong or I am just unable to calculate the maxima. Any help would be appreciated.