20 Find the function with the derivative f x (2024)

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20.Find the function with the derivative $f^{\prime}(x)=e^{6}$ that passes through the point $(0,2)$A. $\frac{e^{6 z}}{6}-\frac{11}{6}$B. $\frac{e^{6}}{6 x}+\frac{11}{6}$C. $6 e^{5}+2$D. $\frac{e^{7}}{7}+2$E. $e^{7 x}$F. $e^{6} x+2$Next item

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#### Solution By Steps***Step 1: Understand the Problem***We need to find a function $ f(x) $ whose derivative is $ f'(x) = e^6 $ and which passes through the point (0, 2).***Step 2: Integrate the Derivative***To find $ f(x) $, integrate $ f'(x) $:\[ f(x) = \int e^6 \, dx \]Since $ e^6 $ is a constant, the integral is:\[ f(x) = e^6 x + C \]where $ C $ is the constant of integration.***Step 3: Use the Given Point to Find $ C $***The function passes through the point (0, 2), so substitute $ x = 0 $ and $ f(x) = 2 $ into the equation:\[ 2 = e^6 \cdot 0 + C \]\[ C = 2 \]***Step 4: Write the Final Function***Substitute $ C $ back into the function:\[ f(x) = e^6 x + 2 \]#### Final Answer\[ \boxed{f(x) = e^6 x + 2} \]The correct option is F.

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20 Find the function with the derivative f x (2024)

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