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Question #40989

Difficulty: 8/10 | 🟢 Free

The Question

Let $x_1, x_2, x_3, x_4$ be real numbers satisfying the following conditions: $\sum_{i=1}^4 \cos x_i = 0, \quad \sum_{i=1}^4 \sin x_i = 0, \quad \sum_{i=1}^4 x_i = 2\pi$ Find the minimum possible value of the objective function $f(x) = \sum_{i=1}^4 x_i^2$ .

Options

A.
$\pi^2$
B.
$2\pi^2$
C.
$\frac{9\pi^2}{4}$
D.
$3\pi^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Variance Invariance under Translation|Objective Function Gradient|Conditional Identity Constraints

Why This Question Is Tricky

Common Mistakes

This problem tests the concept of variance invariance under translation combined with trigonometric identity constraints.

The trap lies in attempting to use Lagrange multipliers directly on $x_i$ , which leads to a complex transcendental equation $x_i - \lambda = R \cos(x_i + \alpha)$ that is difficult to solve without recognizing the periodicity and geometric meaning of the constraints.

By shifting the variables by their mean ( $\pi/2$ ), students can decouple the algebraic objective function from the trigonometric constraints.

A common mistake is assuming the minimum occurs when the angles are equally spaced (forming a square, yielding $2.25\pi^2$ ), ignoring that the vectors can form degenerate polygons (two pairs of identical opposite vectors) which provides the true global minimum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Variance Invariance under Translation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free