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Question #40667

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = 2x + \frac{18}{x}$ for $x > 0$ . Let $x_0$ be the point where $f(x)$ attains its minimum value, and let $m$ be this minimum value. Let $g: (0, \infty) \to \mathbb{R}$ be a continuous function such that $g(1) = -1$ and its derivative is given by: $g'(x) = \begin{cases} \cos(\pi x) - \pi x \sin(\pi x) & \text{for } 0 < x < x_0 \\ (x+1)e^{x-x_0} & \text{for } x > x_0 \end{cases}$ Find the value of $g(m)$ .

Options

A.
$12e^9 - 6$
B.
$12e^9$
C.
$12e^9 - 3$
D.
$12e^9 + 3$

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

Arithmetic Mean Inequality|Integrating f(x) + xf'(x) forms|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Integration Constant Trap' in the context of piecewise functions, combined with the recognition of the exact derivative form $(xf(x))'$ and the AM-GM inequality.

Students must first use AM-GM to find the piecewise boundary $x_0$ and the evaluation point $m$ .

The core trap lies in integrating the second piece of the derivative: students often indefinitely integrate $(x+1)e^{x-3}$ to $xe^{x-3}$ and assume the integration constant is zero, completely ignoring the requirement to enforce continuity at $x=3$ .

This leads directly to the trap answer $12e^9$ .

Proper execution requires matching the left-hand limit $g(3)=-3$ to the right-hand limit to find the hidden constant $C_2 = -6$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Arithmetic Mean Inequality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free