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Question #41097

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable uniformly distributed on the set $\{-10, -9, \dots, 9, 10\}$ . Define the function $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = E[\max(x, X) + \max(-x, -X)]$ Let $S$ be the set of all points $c \in (0, 2)$ such that $f(2) - f(0) = 2f'(c)$ . What is the number of elements in $S$ ?

Options

A.
0
B.
1
C.
2
D.
3

Check Your Answer

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

Symmetry and Invariance|Random Variable Expectation|Mean Value Theorem Applicability

Why This Question Is Tricky

Common Mistakes

This question tests the applicability of the Mean Value Theorem (MVT) on functions defined via expectations of discrete random variables.

Students often incorrectly assume that an expectation function $f(x)$ is everywhere differentiable, especially when disguised using continuous functions like $\max$ .

The trap lies in blindly applying the MVT formula $\frac{f(b)-f(a)}{b-a} = f'(c)$ without verifying differentiability on the open interval.

Here, $f(x)$ simplifies to a piecewise linear function with a 'kink' at $x=1$ , meaning $f'(1)$ is undefined.

The average rate of change over $[0, 2]$ is $2/21$ , but the actual derivative only takes values $1/21$ and $3/21$ .

Thus, the MVT conclusion fails, and no such $c$ exists.

Quick Facts About This Question

Question Information

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