**Correct Option:** D
**Reasoning:**
* \(f(x)=|x|\) has \(f(x) \ge 0\) and \(f(0)=0\), so minimum at \(x=0\).
* \( \lim_{x \to \pm\infty} |x| = \infty \), so no maximum value.
* \( \lim_{x \to 0} |x| = 0 = f(0) \), so continuous at \(x=0\).
* LHD at \(x=0\) is -1, RHD at \(x=0\) is 1. Not differentiable at \(x=0\).
AI Tutor Explanation
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Step-by-Step Solution
Analyze the function \(f(x) = |x|\):
The function \(f(x) = |x|\) is defined as:
\[
f(x) = \begin{cases}
x, & \text{if } x \geq 0 \\
-x, & \text{if } x < 0
\end{cases}
\]
Check for minimum value:
At \(x = 0\), \(f(0) = |0| = 0\). For all other values of \(x\), \(f(x) = |x| > 0\). Thus, \(f\) has a minimum value at \(x = 0\). So, statement (A) is correct.
Check for maximum value:
As \(x\) increases (or decreases), \(|x|\) also increases without bound. Therefore, \(f\) has no maximum value in R. So, statement (B) is correct.
Check for continuity at \(x = 0\):
The left-hand limit is \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x = 0\).
The right-hand limit is \(\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0\).
Since the left-hand limit, right-hand limit, and the function value at \(x = 0\) are all equal to 0, \(f\) is continuous at \(x = 0\). So, statement (C) is correct.
Check for differentiability at \(x = 0\):
The left-hand derivative is \(\lim_{h \to 0^-} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0^-} \frac{|h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1\).
The right-hand derivative is \(\lim_{h \to 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0^+} \frac{|h| - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1\).
Since the left-hand derivative and the right-hand derivative are not equal, \(f\) is not differentiable at \(x = 0\). So, statement (D) is incorrect.
Correct Answer: D
AI Suggestion: Option D
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires students to apply their understanding of absolute value functions, continuity, differentiability, and maximum/minimum values to analyze the given function and identify the incorrect statement.
Knowledge Dimension:CONCEPTUAL
Justification:The question tests the understanding of concepts like continuity, differentiability, and extrema of a function, rather than recalling facts or following a specific algorithm.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding of the properties of functions, continuity, and differentiability, which are core concepts covered in the textbook.