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Question #41624

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be the point that divides the line segment joining the points $A(1, -2, 3)$ and $B(4, 7, -6)$ internally in the ratio $1:2$ . Let $c$ be the perpendicular distance of point $P$ from the $yz$ -plane. Consider the function $f(x) = x^2 - 5x - 3|x - c|$ defined on the interval $[1, 5]$ . If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(x)$ on the given interval respectively, then the value of $M + m$ is:

Options

A.
$-16$
B.
$-17$
C.
$-14$
D.
$-19$

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

Section Formula Internals|Global Extrema on Intervals|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of 3D coordinate geometry and calculus.

The section formula determines a geometric parameter $c$ that acts as the critical boundary for a modulus function.

The primary trap lies in the non-differentiability of the modulus at $x=2$ .

Students who rely solely on setting the derivative to zero will find critical points at $x=1$ and $x=4$ , completely missing the local maximum at the non-differentiable point $x=2$ .

Since $f(2) = -6$ is strictly greater than the boundary values $f(1) = -7$ and $f(5) = -9$ , ignoring the modulus boundary leads to an incorrect absolute maximum and a wrong final sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Section Formula Internals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free