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Question #41666

Difficulty: 8/10 | 🟢 Free

The Question

Let a line $L$ lie in the plane $P: 2x + y + 2z = 12$ . The projection of the line $L$ onto the $yz$ -plane is the line given by the equation $y - 2z = 4$ . Let $S$ be the set of all points $(x,y,z)$ on $L$ that satisfy the non-negativity conditions $x \ge 0$ , $y \ge 0$ , and $z \ge 0$ . If $M$ and $m$ are the absolute maximum and absolute minimum values of the function $f(x,y,z) = x^2 + y^2 + z^2 - 6z - 22$ on the set $S$ , respectively, find the value of $M + m$ .

Options

A.
43
B.
139
C.
67
D.
53

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

Boundary Value Theorem|Non-Negativity Restriction|Line lying in a plane|Coordinate Axis Projections

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle 3D line constraints and coordinate axis projections under non-negativity restrictions.

The core trap lies in the dimensionality reduction during projection: students often analyze the $yz$ -projection ( $y - 2z = 4$ ) and the plane's $y$ -intercept ( $y \le 12$ ) to incorrectly deduce the parameter domain (e.g., $t \in [0, 4]$ ).

However, the 3D non-negativity constraint $x \ge 0$ strictly bounds the line segment to $t \in [0, 2]$ .

Failing to intersect all three half-spaces ( $x \ge 0, y \ge 0, z \ge 0$ ) leads to evaluating the Boundary Value Theorem over an erroneously expanded feasible region, yielding incorrect extrema (such as a maximum of 130 at $t=4$ , leading to the distractor 139).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Boundary Value Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free