The derivative of a function $f$ is given by $f^{\prime }(x)=0.1x+e^{0.25x}$. At what value of $x$ for $x>0$ does the line tangent to the graph of $f$ at $x$ have slope 2 ?
(A) $0.512$
(B) 1.849
(C) $2.287$
(D) 8.113

Step 1 ：The derivative of a function $f$ is given by $f^{\prime }(x)=0.1x+e^{0.25x}$. We are asked to find the value of $x$ for $x>0$ at which the line tangent to the graph of $f$ at $x$ has a slope of 2.

Step 2 ：The slope of the tangent line to the graph of a function at a point is given by the derivative of the function at that point. Therefore, we need to find the value of $x$ for which $f^{\prime }(x)=2$. This leads to the equation $0.1x+e^{0.25x}=2$.

Step 3 ：We can solve this equation numerically. Let's start with an initial guess of $x=1$.

Step 4 ：By solving the equation numerically, we find that the solution to the equation $0.1x+e^{0.25x}=2$ is approximately 2.287.

Step 5 ：This is the value of $x$ for which the slope of the tangent line to the graph of $f$ at $x$ is 2.

Step 6 ：Final Answer: $\boxed{{\displaystyle 2.287}}$