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Question #41864

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_{0}^{3x} |t^2 - 3xt + 2x^2| \,dt$ for $x > 0$ . Find the value of $f'(2)$ .

Options

A.
18
B.
22
C.
24
D.
28

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

Modulus Sign Dependency|Newton-Leibniz Formula with variable limits|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula and definite integration properties when the integrand contains a modulus function whose roots depend on the variable of differentiation.

Students frequently fall into the trap of applying the Leibniz rule blindly without resolving the modulus, or they ignore the modulus entirely, which leads to integrating $t^2 - 3xt + 2x^2$ directly over $[0, 3x]$ .

This incorrect approach yields $f(x) = 3x^3/2$ and $f'(2) = 18$ .

The rigorous method requires splitting the integral at the roots $t=x$ and $t=2x$ before evaluating or differentiating.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Sign Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free