$\begin{array}{l}f^{\prime}(x)=-27 e^{x} \text { and } f(6)=36-27 e^{6} . \\ f(0)=\square\end{array}$

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Find higher derivatives of rational and radical functions

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\begin{array}{l}f^{\prime}(x)=-27 e^{x} \text { and } f(6)=36-27 e^{6} . \\ f(0)=\square\end{array}

Find Antiderivative: To find $f(0)$ , we need to integrate the derivative $f'(x) = -27e^x$ to get the original function $f(x)$ . The antiderivative of $-27e^x$ is $-27e^x$ , since the derivative of $e^x$ is $e^x$ and the constant multiple rule of integration allows us to pull out the $-27$ .
Add Constant C: After finding the antiderivative, we add a constant C to represent the indefinite integral: $f(x) = -27e^x + C$ .
Use Given Information: We are given that $f(6) = 36 - 27e^6$ . We can use this information to solve for the constant $C$ . We substitute $x$ with $6$ in the antiderivative: $f(6) = -27e^6 + C$ .
Solve for Constant C: Now we set the equation equal to the given value of $f(6)$ : $-27e^6 + C = 36 - 27e^6$ .
Substitute Values: Solving for $C$ , we add $27e^6$ to both sides of the equation: $C = 36 - 27e^6 + 27e^6$ .
Calculate Constant C: Simplifying the right side of the equation, we find that $C = 36$ , since $-27e^6 + 27e^6$ cancels out.
Write Complete Function: Now that we have the value of $C$ , we can write the complete function $f(x)$ : $f(x) = -27e^x + 36$ .
Substitute $x$ with $0$ : To find $f(0)$ , we substitute $x$ with $0$ in the function $f(x)$ : $f(0) = -27e^0 + 36$ .
Simplify Equation: Since $e^0$ is equal to $1$ , the equation simplifies to: $f(0) = -27(1) + 36$ .
Calculate $f(0)$ : Calculating the value, we get $f(0) = -27 + 36$ .
Calculate $f(0)$ : Calculating the value, we get $f(0) = -27 + 36$ .Finally, we find that $f(0) = 9$ .