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Question #40363

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [0, 4] \to \mathbb{R}$ be defined as $f(x) = \text{sgn}(x) \cdot \left( |x^2 - 4x + 3| + 2 \right)$ where $\text{sgn}(x)$ denotes the signum function. If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(x)$ on the interval $[0, 4]$ respectively, then the value of $M + m$ is:

Options

A.
5
B.
7
C.
0
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

Signum Function Discontinuity|Global Extrema on Intervals|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of global extrema when standard derivative tests fail due to discontinuities and cusps.

The function features non-differentiable cusps at $x=1$ and $x=3$ , rendering the second derivative test inapplicable, and a jump discontinuity at $x=0$ .

A common cognitive trap is to blindly apply calculus techniques to the continuous portion of the function while ignoring the isolated point created by the signum function.

Students will correctly find the local minimum of 2 at the cusps but will incorrectly assume it is the global minimum, missing the true global minimum of 0 hidden precisely at the signum jump point $x=0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Signum Function Discontinuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free