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Question #40507

Difficulty: 8/10 | 🟢 Free

The Question

Let $a$ be the minimum value of the function $P(x) = x^2 - 2x + 2$ and $b$ be the maximum value of the function $Q(x) = -x^2 + 10x - 20$ . Consider the function $F(x)$ defined on $[a, b]$ by $F(x) = \int_{a}^{x} \ln|t^2 - 4t + 3| \, dt$ Let $S$ be the set of all points $x \in (a, b)$ such that $F'(x) = 0$ . Find the sum of all elements in $S$ .

Options

A.
$4 + \sqrt{2}$
B.
$2 + \sqrt{2}$
C.
$4$
D.
$2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Maximum/Minimum of Quadratic Expressions|Newton-Leibniz Formula with variable limits|Logarithmic Integration Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula combined with the 'Logarithmic Integration Modulus' trap.

Students must first determine the integration interval by finding the extrema of two quadratics.

The core trap lies in solving $F'(x) = \ln|x^2 - 4x + 3| = 0$ .

A common mistake is to ignore the absolute value, solving only $x^2 - 4x + 3 = 1$ and missing the root derived from $x^2 - 4x + 3 = -1$ .

Because the vertex of the quadratic inside the logarithm drops below zero, the modulus is activated, making $x=2$ a valid critical point where the derivative is zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Maximum/Minimum of Quadratic Expressions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free