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Question #41045

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a continuous random variable with probability density function $f(x) = \begin{cases} a x^2 + b x + c & \text{for } x \in [-1, 1] \\ 0 & \text{otherwise} \end{cases}$ , where $a, b, c$ are real constants. What is the maximum possible value of the expected value $E[X]$ ?

Options

A.
$\frac{1}{\sqrt{3}}$
B.
$\frac{1}{2}$
C.
$\frac{1}{3}$
D.
$\frac{\sqrt{3}}{2}$

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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

Axiomatic Probability Constraints|Inequalities in Definite Integrals|Closed vs Open Intervals|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of Axiomatic Probability Constraints ( $f(x) \ge 0, \int f = 1$ ) and Global Extrema on closed intervals.

The trap lies in the endpoint behavior: students often only check the boundary conditions $f(1) \ge 0$ and $f(-1) \ge 0$ , which incorrectly suggests $b$ can grow indefinitely if $a$ is large, or yields a suboptimal maximum of $1/2$ if they misapply the vertex constraint.

To find the true maximum, one must rigorously evaluate the global minimum of the quadratic within the closed interval, ensuring the vertex condition $x_v \in [-1, 1]$ is satisfied when the discriminant inequality is optimized.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Axiomatic Probability Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free