Maths · Limits, continuity and differentiability
Consider the functions, =\mid \boldsymbol{x} \) Statement 1: =\mathbf{0} \) Stat
Consider the functions, \( \boldsymbol{f}(\boldsymbol{x})=\mid \boldsymbol{x} \) \( \mathbf{2}|+| \boldsymbol{x}-\mathbf{5} \mid, \boldsymbol{x} \in \boldsymbol{R} \) Statement 1: \( \boldsymbol{f}^{\prime}(\mathbf{4})=\mathbf{0} \) Statement \( 2: f \) is continuous in [2,5] differentiable in (2,5) and \( f(2)=f(5) \)
- A. Statement 1 is false, Statement 2 is true
- B. Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement
- C. statement 1 is true, statement 2 is true; Statement 2 is not a correct explanation for Statement 1
- D. Statement 1 is true, Statement 2 is false
Step-by-step solution
f(x)=|x-2|+|x-5|. On (2,5), f(x)=3 constant, so f'(4)=0 (Statement 1 true). f is continuous on [2,5], differentiable on (2,5), and f(2)=f(5)=3 (Statement 2 true). Rolle's theorem guarantees some c in (2,5) with f'(c)=0, but not necessarily c=4; the fact that f'(4)=0 follows from constancy, not directly from the conditions. Hence Statement 2 is not a correct explanation for Statement 1.
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