Step 1: Describe the random variable.
The random variable $X$ can take only two possible values: $1$ and $2$.
Let the probabilities be defined as:
\[ P(X=1)=p \quad \text{and} \quad P(X=2)=1-p \]
The expected value of $X$ is given as $E[X]=\frac{10}{7}$.
Using the definition of expectation:
\[ E[X] = 1\cdot p + 2\cdot(1-p) = 2 - p \]
Equating this to the given mean:
\[ 2 - p = \frac{10}{7} \]
Solving for $p$:
\[ p = 2 - \frac{10}{7} = \frac{4}{7} \]
Hence,
\[ P(X=1)=\frac{4}{7}, \quad P(X=2)=\frac{3}{7} \]
Step 2: Construct the moment generating function.
The moment generating function (MGF) of $X$ is defined as:
\[ M_X(t)=E[e^{tX}] \]
Substituting the values of $X$ and their probabilities:
\[ M_X(t)=\frac{4}{7}e^{t}+\frac{3}{7}e^{2t} \]
Step 3: Differentiate the MGF four times.
Taking successive derivatives:
\[ M_X'(t)=\frac{4}{7}e^{t}+\frac{6}{7}e^{2t} \]
\[ M_X''(t)=\frac{4}{7}e^{t}+\frac{12}{7}e^{2t} \]
\[ M_X^{(3)}(t)=\frac{4}{7}e^{t}+\frac{24}{7}e^{2t} \]
\[ M_X^{(4)}(t)=\frac{4}{7}e^{t}+\frac{48}{7}e^{2t} \]
Now evaluate the fourth derivative at $t=0$:
\[ M_X^{(4)}(0)=\frac{4}{7}+\frac{48}{7}=\frac{52}{7} \]
Step 4: Final result.
The value of the fourth derivative of the moment generating function at zero is:
\[ \boxed{\frac{52}{7}} \]