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Question #41363

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x (t^2 - 3t + 2) \,dt$ and $g(x) = x^2 - 6x + 8$ . If $F(x) = f(g(x))$ , then the set of all values of $x$ for which $F(x)$ is strictly increasing is:

Options

A.
$(3-\sqrt{3}, 3-\sqrt{2}) \cup (3, 3+\sqrt{2}) \cup (3+\sqrt{3}, \infty)$
B.
$(-\infty, 3-\sqrt{3}) \cup (3-\sqrt{2}, 3) \cup (3+\sqrt{2}, 3+\sqrt{3})$
C.
$(3-\sqrt{3}, 3-\sqrt{2}) \cup (3+\sqrt{2}, 3+\sqrt{3})$
D.
$(-\infty, 3-\sqrt{2}) \cup (3, 3+\sqrt{2}) \cup (3+\sqrt{3}, \infty)$

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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

Increasing/Decreasing Strictness|Maximum/Minimum of Quadratic Expressions|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the monotonicity of a composite function $F(x) = f(g(x))$ using the chain rule $F'(x) = f'(g(x))g'(x)$ .

A common trap is analyzing only $f'(g(x)) > 0$ and ignoring the sign change of the inner function's derivative $g'(x)$ at its vertex.

By nesting a quadratic inside a function with multiple roots, the problem creates compound boundaries where the global behavior of the quadratic interacts with the local roots of $f'(x)$ .

Students must perform a rigorous multi-interval sign analysis (wavy curve method) across five critical points, correctly identifying that the monotonicity of $F(x)$ inverts whenever $g(x)$ is decreasing.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Increasing/Decreasing Strictness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free