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Question #41852

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^{\sin x} \left(t^2 - \frac{1}{4}\right)^3 \left(t^2 - \frac{3}{4}\right)^2 dt$ . Let $S$ be the set of all critical points of $f(x)$ in the interval $(0, 2\pi)$ . Let $m$ , $M$ , and $I$ be the number of points in $S$ at which $f(x)$ attains a local minimum, a local maximum, and a point of inflection, respectively. Three independent events $E_1, E_2$ , and $E_3$ have probabilities $P(E_1) = \frac{m}{m+M+I}$ , $P(E_2) = \frac{M}{m+M+I}$ , and $P(E_3) = \frac{I}{m+M+I}$ . Find the probability that exactly two of the events $E_1, E_2, E_3$ occur.

Options

A.
$\frac{111}{500}$
B.
$\frac{1}{4}$
C.
$\frac{27}{125}$
D.
$\frac{113}{500}$

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

Multiplication Principle Scope|Substitution (Change of Variable)|Second Derivative Test Limitations|Point of Inflection Conditions

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Calculus (First/Second Derivative Tests) and Probability (Multiplication Principle).

The primary trap lies in the evaluation of the critical points of the integral function.

Students often blindly apply the Second Derivative Test or assume all roots of $f'(x)=0$ are local extrema.

Because the integrand contains a squared factor $(t^2 - 3/4)^2$ , the derivative $f'(x)$ does not change sign at these roots, creating stationary points of inflection.

Failing to identify these inflection points leads to incorrect values for $m, M,$ and $I$ , which cascades into the probability calculation, invalidating the multiplication of the independent events.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Multiplication Principle Scope
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free