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Question #41135

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = x + \frac{1}{x}$ and $g(x) = e^x$ . Define the integrals $I_1 = \int_{1/2}^{2} f(g(x)) \, dx$ and $I_2 = \int_{1/2}^{2} g(f(x)) \, dx$ . Which of the following inequalities is correct?

Options

A.
$I_1 < e^2$ and $I_2 > \frac{3}{2}e^2$
B.
$I_1 > e^2$ and $I_2 > \frac{3}{2}e^2$
C.
$I_1 < e^2$ and $I_2 < \frac{3}{2}e^2$
D.
$I_1 > e^2$ and $I_2 < \frac{3}{2}e^2$

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

Composition Non-Commutativity|Arithmetic Mean Inequality|Inequalities in Definite Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of function composition alongside the application of the AM-GM inequality within definite integrals.

A common trap is for students to incorrectly evaluate $I_1$ by neglecting the lower bound of integration (assuming $\int e^x \, dx \approx e^2$ and adding the positive $\int e^{-x} \, dx$ , leading to the false conclusion $I_1 > e^2$ ).

Furthermore, students often misapply or confuse the bounds of $f(g(x))$ and $g(f(x))$ , failing to recognize that the inner function dictates the domain for the outer function's inequality.

Here, AM-GM elegantly provides a strict lower bound for $I_2$ , while precise algebraic evaluation bounds $I_1$ from above.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Composition Non-Commutativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free