The derivative of a function $f$ is given by $f^{\prime}(x)=0.1 x+e^{0.25 x}$. At what value of $x$ for $x>0$ does the line tangent to the graph of $f$ at $x$ have slope 2 ? (A) $\quad 0.512$ (B) 1.849 (C) $\quad 2.287$ (D) 8.113
Solution
Step 1 ：The derivative of a function \(f\) is given by \(f^{\prime}(x)=0.1 x+e^{0.25 x}\). 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 \(0.1 x+e^{0.25 x}=2\).
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 \(0.1 x+e^{0.25 x}=2\) 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: \(\boxed{2.287}\)
