Key Concepts and Formulas

1. Differentiation of Exponential Functions: The derivative of $ae^{kx}$ is $ake^{kx}$.
2. Properties of Logarithms and Exponentials: $e^{\log_e x} = x$ and $e^{n \log_e x} = x^n$. Also, $e^0 = 1$.
3. Definite Integration: The definite integral of a function $g(x)$ from $a$ to $b$ is found by evaluating its antiderivative $G(x)$ at the limits: $\int_a^b g(x) dx = G(b) - G(a)$.
4. Solving Systems of Linear Equations: Using algebraic manipulation to find the values of unknown variables.

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Step-by-Step Solution:

We are given the function $f(x) = a e^{2x} + b e^x + c x$ and three conditions that will allow us to determine the values of the constants $a$, $b$, and $c$.

Step 1: Use the condition $f(0) = -1$ to form the first equation.

We are given $f(x) = a e^{2x} + b e^x + c x$. Substitute $x=0$ into the function: $f(0) = a e^{2(0)} + b e^0 + c(0)$ $f(0) = a e^0 + b e^0 + 0$ Using the property $e^0 = 1$: $f(0) = a(1) + b(1)$ $f(0) = a + b$ We are given that $f(0) = -1$. Therefore, our first equation is: $a + b = -1 \quad (\text{Equation 1})$ Explanation: This step utilizes the definition of a function and the property of the exponential function at zero to establish a linear relationship between the constants $a$ and $b$.

Step 2: Use the condition $f'(\log_e 2) = 21$ to form the second equation.

First, we need to find the derivative of $f(x)$, denoted by $f'(x)$. $f(x) = a e^{2x} + b e^x + c x$ Differentiating term by term with respect to $x$:

• The derivative of $a e^{2x}$ is $a \cdot (2e^{2x}) = 2a e^{2x}$.
• The derivative of $b e^x$ is $b e^x$.
• The derivative of $c x$ is $c$. So, the derivative function is: $f'(x) = 2a e^{2x} + b e^x + c$ Now, we substitute $x = \log_e 2$ into $f'(x)$: $f'(\log_e 2) = 2a e^{2(\log_e 2)} + b e^{\log_e 2} + c$ Using the properties $e^{n \log_e x} = x^n$ and $e^{\log_e x} = x$:
• $e^{2(\log_e 2)} = e^{\log_e (2^2)} = e^{\log_e 4} = 4$.
• $e^{\log_e 2} = 2$. Substituting these values: $f'(\log_e 2) = 2a (4) + b (2) + c$ $f'(\log_e 2) = 8a + 2b + c$ We are given that $f'(\log_e 2) = 21$. Therefore, our second equation is: $8a + 2b + c = 21 \quad (\text{Equation 2})$ Explanation: This step involves differentiating the function and then applying the given condition using properties of logarithms and exponentials to establish another linear relationship between $a$, $b$, and $c$.

Step 3: Use the condition $\int_0^{\log_e 4}(f(x)-c x) d x=\frac{39}{2}$ to form the third equation.

First, let's find the expression for $f(x) - cx$: $f(x) - cx = (a e^{2x} + b e^x + c x) - cx$ $f(x) - cx = a e^{2x} + b e^x$ Now, we need to evaluate the definite integral of this expression from $0$ to $\log_e 4$: $\int_0^{\log_e 4} (a e^{2x} + b e^x) dx$ Let's find the antiderivative of $(a e^{2x} + b e^x)$:

• The antiderivative of $a e^{2x}$ is $a \left(\frac{e^{2x}}{2}\right) = \frac{a}{2} e^{2x}$.
• The antiderivative of $b e^x$ is $b e^x$. So, the antiderivative of $(a e^{2x} + b e^x)$ is $\frac{a}{2} e^{2x} + b e^x$. Now, we apply the Fundamental Theorem of Calculus: $\left[\frac{a}{2} e^{2x} + b e^x\right]_0^{\log_e 4} = \frac{39}{2}$ Evaluate the expression at the upper limit ($x = \log_e 4$) and subtract the evaluation at the lower limit ($x = 0$): At $x = \log_e 4$: $\frac{a}{2} e^{2(\log_e 4)} + b e^{\log_e 4} = \frac{a}{2} e^{\log_e (4^2)} + b(4) = \frac{a}{2} e^{\log_e 16} + 4b = \frac{a}{2}(16) + 4b = 8a + 4b$ At $x = 0$: $\frac{a}{2} e^{2(0)} + b e^0 = \frac{a}{2} e^0 + b e^0 = \frac{a}{2}(1) + b(1) = \frac{a}{2} + b$ Now, subtract the lower limit evaluation from the upper limit evaluation: $(8a + 4b) - \left(\frac{a}{2} + b\right) = \frac{39}{2}$ $8a + 4b - \frac{a}{2} - b = \frac{39}{2}$ Combine like terms: $(8a - \frac{a}{2}) + (4b - b) = \frac{39}{2}$ $(\frac{16a - a}{2}) + 3b = \frac{39}{2}$ $\frac{15a}{2} + 3b = \frac{39}{2}$ To eliminate the fractions, multiply the entire equation by 2: $15a + 6b = 39$ We can simplify this equation by dividing by 3: $5a + 2b = 13 \quad (\text{Equation 3})$ Explanation: This step involves simplifying the integrand, finding its antiderivative, and then applying the definite integral formula by evaluating at the limits. Properties of logarithms and exponentials are crucial here, as is algebraic simplification to obtain a linear equation in terms of $a$ and $b$.

Step 4: Solve the system of linear equations to find the values of $a$, $b$, and $c$.

We have the following system of equations:

1. $a + b = -1$
2. $8a + 2b + c = 21$
3. $5a + 2b = 13$

Let's first solve for $a$ and $b$ using Equations 1 and 3. From Equation 1, we can express $b$ in terms of $a$: $b = -1 - a$ Substitute this expression for $b$ into Equation 3: $5a + 2(-1 - a) = 13$ $5a - 2 - 2a = 13$ $3a - 2 = 13$ $3a = 15$ $a = 5$ Now, substitute the value of $a$ back into the expression for $b$: $b = -1 - a = -1 - 5$ $b = -6$ Now that we have the values of $a$ and $b$, we can substitute them into Equation 2 to find $c$: $8a + 2b + c = 21$ $8(5) + 2(-6) + c = 21$ $40 - 12 + c = 21$ $28 + c = 21$ $c = 21 - 28$ $c = -7$ So, the values of the constants are $a=5$, $b=-6$, and $c=-7$. Explanation: This step involves a standard procedure for solving a system of linear equations. We used substitution to solve for two variables first and then used those values to find the third variable.

Step 5: Calculate $|a+b+c|$.

We have found $a=5$, $b=-6$, and $c=-7$. Now, we calculate the sum $a+b+c$: $a+b+c = 5 + (-6) + (-7)$ $a+b+c = 5 - 6 - 7$ $a+b+c = -1 - 7$ $a+b+c = -8$ Finally, we need to find the absolute value: $|a+b+c| = |-8|$ $|a+b+c| = 8$ Explanation: This final step involves substituting the determined values of the constants into the expression $|a+b+c|$ and computing its absolute value.

Common Mistakes & Tips

• Logarithm and Exponential Properties: Ensure correct application of rules like $e^{\log_e x} = x$ and $e^{n \log_e x} = x^n$. Mistakes here can lead to incorrect values in Steps 2 and 3.
• Algebraic Errors: Be meticulous with algebraic manipulations, especially when solving systems of equations or simplifying expressions with fractions. Double-checking calculations can prevent errors.
• Differentiation/Integration Errors: Carefully apply the rules of differentiation and integration for exponential functions. The constants of integration are not needed for definite integrals.

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Summary

The problem was solved by systematically using the three given conditions to form a system of linear equations for the unknown constants $a$, $b$, and $c$. The first condition, $f(0)=-1$, provided a direct relation between $a$ and $b$. The second condition, involving the derivative $f'(\log_e 2)=21$, yielded another linear equation in $a$, $b$, and $c$. The third condition, a definite integral, was evaluated to produce a third linear equation involving $a$ and $b$. Solving this system of equations yielded $a=5$, $b=-6$, and $c=-7$. Finally, the absolute value of the sum of these constants, $|a+b+c|$, was calculated to be 8.

The final answer is $\boxed{8}$.