{:[f^(')(x)=-27e^(x)" and "],[f(6)=36-27e^(6).],[f(0)=◻]:}

Question

Bytelearn · Accepted Answer

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.Finally, we find that f(0) = 9.

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

Full solution

More problems from Find higher derivatives of rational and radical functions

Related topics

f′(x)=−27ex and f(6)=36−27e6.f(0)=□ \begin{array}{l}f^{\prime}(x)=-27 e^{x} \text { and } f(6)=36-27 e^{6} . \\ f(0)=\square\end{array} f′(x)=−27ex and f(6)=36−27e6.f(0)=□​

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