Before You Begin

Question #41083

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = 3x^4 - 16x^3 + 24x^2$ be a function defined on the sample space $\Omega = [-1, 3]$ . A continuous random variable $X$ takes values in $\Omega$ with a probability density function $p(x) = \frac{x+1}{8}$ . Let $x_1$ and $x_2$ be the points in $\Omega$ where $f(x)$ attains its global minimum and global maximum, respectively. Let $A$ be the event that $X \in [\min(x_1, x_2), \max(x_1, x_2)]$ , and $B$ be the event that $X$ falls in the interval where the graph of $f(x)$ is strictly concave downward. Calculate the probability $P(A' \cap B')$ .

Options

A.
$\frac{79}{144}$
B.
$\frac{65}{144}$
C.
$\frac{1}{2}$
D.
$\frac{1}{16}$

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

De Morgan’s Laws in Probability|Global Extrema on Intervals|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to synthesize global extrema analysis, concavity, and De Morgan's laws within a continuous probability framework.

The primary trap lies in the evaluation of the global maximum.

Students often rely exclusively on the Second Derivative Test to classify extrema.

For $f(x)$ , $f''(2) = 0$ , making the test inconclusive.

A common failure mode is incorrectly assuming $x=2$ is a local/global maximum without checking the boundaries of $\Omega = [-1, 3]$ .

If a student neglects the boundary $x=-1$ (where the true global maximum $f(-1)=43$ occurs) and assumes the maximum is at $x=2$ , they incorrectly define Event $A$ as $[0, 2]$ .

This leads to $P(A) = 1/2$ and a cascading error resulting in the trap answer $1/2$ .

Furthermore, the disjoint nature of the correct intervals requires careful application of De Morgan's law $P(A' \cap B') = 1 - P(A \cup B)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	De Morgan’s Laws in Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free