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Question #41378

Difficulty: 8/10 | 🟢 Free

The Question

Let $F(x) = \int_{0}^{x} (t^2 - 5t + 4) \ln(3 - t) \, dt$ be defined on its maximal domain $(-\infty, 3)$ . If $x_0$ is the point of local maximum of $F(x)$ , then the value of $F(x_0)$ is:

Options

A.
$\frac{10}{3} \ln 2 - \frac{3}{2} \ln 3 + \frac{41}{36}$
B.
$-\frac{3}{2} \ln 3 + \frac{28}{9}$
C.
$\frac{10}{3} \ln 2 + \frac{3}{2} \ln 3 - \frac{41}{36}$
D.
$\frac{5}{3} \ln 2 - \frac{1}{2} \ln 3 + \frac{17}{18}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Global Extrema on Intervals|Integration by Parts (LIATE Rule)|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the 'Compound Function Boundary' trap combined with the Fundamental Theorem of Calculus and Integration by Parts (LIATE).

Students often blindly set $F'(x) = 0$ to find critical points, yielding $x=1, 2, 4$ .

The trap lies in ignoring the domain of the integrand ( $t < 3$ ), which restricts the domain of $F(x)$ to $(-\infty, 3)$ .

Failing to reject $x=4$ leads to incorrect sign analysis and misidentification of extrema.

Furthermore, evaluating the integral at the correct local maximum ( $x=1$ ) requires a substitution followed by a rigorous application of the LIATE rule, testing their ability to execute complex definite integrals accurately.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Global Extrema on Intervals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free