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Question #40547

Difficulty: 8/10 | 🟢 Free

The Question

Find the complete set of values of $x$ for which the function $f(x) = \frac{x^2 + 7x - 9}{x^2 + 6x - 7}$ is monotonically increasing.

Options

A.
[-1, 5]
B.
[-1, 1) \cup (1, 5]
C.
(-1, 1) \cup (1, 5)
D.
$(-\infty, -1] \cup [5, \infty)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Cross Multiplication Trap|Maximum/Minimum of Quadratic Expressions|Increasing/Decreasing Strictness

Why This Question Is Tricky

Common Mistakes

This question tests the 'Cross Multiplication Trap' in the context of rational function monotonicity.

Students often compute the derivative $f'(x) = \frac{N(x)}{D(x)^2}$ and, seeing that the denominator is a perfect square, immediately cross-multiply to solve $N(x) \ge 0$ .

This leads them directly to the closed interval $[-1, 5]$ (Option A).

However, they fail to intersect this result with the domain of the original function.

The denominator of $f(x)$ vanishes at $x=1$ , which lies inside the interval $[-1, 5]$ .

Thus, $f(x)$ is discontinuous and non-differentiable at $x=1$ , requiring the interval to be split.

This highlights the critical JEE Advanced requirement of always checking domain constraints before clearing denominators in inequalities.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Cross Multiplication Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free