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Question #41147

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(t) = \sqrt{1 - |t-1|}$ and $F(x) = \int_{x}^{2x} f(t) dt$ . Let $M$ be the maximum value of $F(x)$ on its domain. Then the value of $9M$ is:

Options

A.
$12 - 4\sqrt{2}$
B.
$12 - 2\sqrt{2}$
C.
$6$
D.
$8\sqrt{2} - 4$

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

Domain Restrictions|Newton-Leibniz Formula with variable limits|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula under strict domain restrictions and the 'Modulus Trap'.

Students often blindly differentiate the integral without first establishing the domain of F(x), which is restricted by the domain of the integrand f(t).

Furthermore, the modulus inside the integrand forces the derivative F'(x) to be piecewise.

A common lethal mistake is assuming |2x-1| = 2x-1 globally, which leads to incorrect critical points (like x=6/7).

The student must carefully split the modulus based on the critical point x=1/2 to find the true maximum at x=8/9, and then correctly split the definite integral at t=1 to evaluate the maximum value.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Domain Restrictions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free