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

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:

\(15a + 6b = 39\)

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

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

\(b = -6\)

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

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

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