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Question #41396

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a continuous random variable with probability density function $f(x) = \begin{cases} k(x^2 - ax + b)e^{-x} & x \ge 0 \\ 0 & x < 0 \end{cases}$ where $a, b \in \mathbb{R}$ and $k > 0$ . It is given that the expected value of $X$ is $1$ . The curve $y = f(x)$ has no inflection points on $(0, \infty)$ , yet there exists a unique $c \in (0, \infty)$ such that $f''(c) = 0$ . Find the value of $a + b + c + \frac{1}{k}$ .

Options

A.
14
B.
16
C.
18
D.
20

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

Concavity Analysis|Probability Mass Function (PMF)|Unbounded Region Trap

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between roots of the second derivative and actual inflection points.

A common trap is assuming $f''(c) = 0$ guarantees an inflection point.

Here, the unbounded domain $[0, \infty)$ allows the probability density function to decay to zero while remaining entirely convex ( $f''(x) \ge 0$ ).

This forces the quadratic factor of $f''(x)$ to be a perfect square, creating a double root that does not change the sign of concavity.

It elegantly combines probability constraints (Gamma integrals for total probability and expectation) with differential calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Concavity Analysis
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free