Question

Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be defined \( f(x) = ae^{2x} + be^x + cx \). If \( f(0) = -1 \), \( f'(\log_e 2) = 21 \) and
\[\int_{0}^{\log_e 4} (f(x) - cx) \, dx = \frac{39}{2}\]
then the value of \( |a + b + c| \) equals:

• A. 16
• B. 10
• C. 12
• D. 8

Answer 1

The Correct Option is D

To solve the problem, we need to determine the value of \( |a + b + c| \). We are given a function \( f(x) = ae^{2x} + be^x + cx \) and specific conditions that the function must satisfy.

1. First, find the value of \( f(0) \):
   • Given \( f(0) = -1 \), we substitute \( x = 0 \) into the function: \(f(0) = a \cdot e^0 + b \cdot e^0 + c \cdot 0 = a + b = -1\).
2. Calculate \( f'(\log_e 2) = 21 \):
   • First, find the derivative \( f'(x) \) using \(f(x) = ae^{2x} + be^x + cx\): \(f'(x) = 2ae^{2x} + be^x + c\).
   • Set \( x = \log_e 2 \) in the derivative: \[ f'(\log_e 2) = 2a e^{2 \log_e 2} + b e^{\log_e 2} + c = 8a + 2b + c = 21 \]
3. Evaluate the integral: \[\int_{0}^{\log_e 4} (f(x) - cx) \, dx = \frac{39}{2}\]
   • Consider the expression under the integral, \( f(x) - cx = ae^{2x} + be^x \).
   • Now, integrate: \[ \int_{0}^{\log_e 4} (ae^{2x} + be^x) \, dx = \frac{39}{2} \]
   • The integral evaluates as: \[ \left[\frac{a}{2} e^{2x} + b e^x \right]_{0}^{\log_e 4} = \frac{39}{2} \]
   • Compute: \[ \frac{a}{2} e^{2 \log_e 4} + b e^{\log_e 4} - \left(\frac{a}{2} \cdot e^0 + b \cdot e^0\right) = \frac{39}{2} \]
   • Simplify the exponents: \[ 8a + 4b - \left(\frac{a}{2} + b\right) = \frac{39}{2} \]
   • Further simplification yield: \[ \frac{16a + 8b - a - 2b}{2} = \frac{39}{2} \]
   • Thus: \[ \frac{15a + 6b}{2} = \frac{39}{2} \implies 15a + 6b = 39 \]
4. Now, solve the system of equations:
   • From step 1: \( a + b = -1 \)
   • From step 2: \( 8a + 2b + c = 21 \)
   • From step 3: \( 15a + 6b = 39 \)
5. Substituting \( b = -1 - a \) into the other equations:
   • From step 3, \(15a + 6(-1 - a) = 39\Rightarrow 9a = 45 \Rightarrow a = 5\).
   • From step 1, \(b = -1 - 5 = -6\).
   • Substitute into the step 2 equation: \[ 8(5) + 2(-6) + c = 21 \implies 40 - 12 + c = 21 \implies c = -7 \]
6. Finally, calculate \( |a + b + c| \):
   • \[ a + b + c = 5 - 6 - 7 = -8 \implies |a + b + c| = 8 \]

Therefore, the answer is 8.

Answer 2

The function is given by:

\(f(x) = ae^{2x} + be^x + c\).

Given \(f(0) = -1\):

\(f(0) = a + b + c = -1\).

Differentiate \(f(x)\):

\(f'(x) = 2ae^{2x} + be^x\).

Given \(f'( \log_e 2) = 21\):

\(8a + 2b = 21 \Rightarrow 4a + b = 10.5\).

Evaluate the integral:

\(\int_{0}^{\log_e 4} (f(x) - cx) dx = \int_{0}^{\log_e 4} (ae^{2x} + be^x + c - cx) dx = \frac{39}{2}\).

Break into parts and evaluate each:

\(\frac{15a}{2} + 3b + c \log_e 4 - c \times \frac{(\log_e 4)^2}{2} = \frac{39}{2}\).

Solve for \(a\), \(b\), and \(c\). The value of \(|a + b + c|\) is:

\(15a + 6b = 39\)

\(15a - 6a - 6 = 39\)

\(9a = 45 \Rightarrow a = 5\)

\(b = -6\)

\(c = 21 - 40 + 12 = -7\)

\(a + b + c = -8\)

\(|a + b + c| = 8\)