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Question #40738

Difficulty: 8/10 | 🟢 Free

The Question

Let $\Pi$ be the family of planes passing through the line of intersection of the planes $x = 9$ and $y = 6$ . Let $P_\lambda$ be a plane in $\Pi$ whose equation is given by $x + \lambda y - (9+6\lambda) = 0$ for $\lambda \ge 0$ . Let $d(\lambda)$ be the perpendicular distance from the origin to the plane $P_\lambda$ . Consider the function $f(\lambda) = [d(\lambda)] - 2\lambda$ , where $[ \cdot ]$ denotes the greatest integer function. The global maximum value of $f(\lambda)$ on the interval $[0, 2]$ is:

Options

A.
$\frac{133 + 5\sqrt{17}}{16}$
B.
$\frac{26}{3}$
C.
9
D.
$\frac{117 - 5\sqrt{17}}{16}$

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

General Equation Transformations|Equation of plane through intersection|Greatest Integer Function (GIF)|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between continuous optimization and piecewise constant functions (Greatest Integer Function).

A common trap is assuming the global maximum of $f(\lambda) = [d(\lambda)] - 2\lambda$ occurs where $d(\lambda)$ is maximized (at $\lambda = 2/3$ ).

However, because $[d(\lambda)]$ is constant on intervals, the term $-2\lambda$ forces the maximum to occur at the left boundary of the interval where $[d(\lambda)]$ attains its highest step.

Students must find the exact transition point where $d(\lambda)$ crosses the integer threshold 10, requiring them to solve $d(\lambda) = 10$ and evaluate the function at this non-differentiable jump discontinuity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	General Equation Transformations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free