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Question #40988

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function such that $f(2) = 1$ , $f'(2) = -1$ , $f(4) = 2$ , and $f'(4) = 0$ . A dataset consists of two subgroups, $A$ and $B$ , with constant sizes $n_A = 2$ and $n_B = 3$ respectively. The mean of subgroup $A$ at time $t$ is given by $\mu_A(t) = \int_{1}^{t} (f(x) + xf'(x)) \,dx$ , and the mean of subgroup $B$ at time $t$ is given by $\mu_B(t) = \int_{t}^{t^2} (f(x) + xf'(x)) \,dx$ . If $\mu(t)$ represents the combined mean of the entire dataset at time $t$ , find the value of $\frac{d\mu}{dt}$ at $t = 2$ .

Options

A.
5
B.
$\frac{7}{5}$
C.
4
D.
8

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

Combined Mean of Subgroups|Integrating f(x) + xf'(x) forms|Newton-Leibniz Formula with variable limits

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula (Leibniz Integral Rule) within the context of statistical combined means.

The primary trap lies in differentiating the definite integral with variable upper and lower limits, specifically for $\mu_B(t)$ .

Students frequently forget to apply the chain rule to the upper limit $t^2$ , erroneously evaluating the derivative as $g(t^2) - g(t)$ instead of $g(t^2) \cdot 2t - g(t)$ .

This specific omission leads directly to the incorrect option 7/5.

Additionally, the problem ensures candidates understand that the combined mean of dynamic subgroups is itself a function whose derivative is the weighted sum of the individual derivatives, preventing them from treating the integrals as static constants.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Combined Mean of Subgroups
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free