Consider the following statements in respect of the function f(x)=\begin{cases}|x|+1,&0\lt|x|\le3\\ 1,&x=0\end{cases} 1. The function attains <strong>MAXIMUM</strong> value <strong>ONLY</strong> at x=3 2. The function attains local <strong>MINIMUM</strong> <strong>ONLY</strong> at x=0 Which of the statements given above is/are correct?
- A. 1 <strong>ONLY</strong>
- B. 2 <strong>ONLY</strong> ✓
- C. <strong>BOTH</strong> 1 and 2
- D. <strong>NEITHER</strong> 1 nor 2
Correct Answer: B. 2 <strong>ONLY</strong>
Explanation
The function f(x) = |x|+1 for x \in [-3, 3] \setminus \{0\} and f(0)=1. The maximum value is f(\pm 3) = 4. Thus, it attains maximum at both x=3 and x=-3, making Statement 1 false. At x=0, f(0)=1, and for any other x in the domain, f(x) \gt 1. Therefore, x=0 is the unique local and absolute minimum. Statement 2 is true.
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