Question:
The function \(f(x)=|x+2|\) is not differentiable at a point
The function \(f(x)=|x+2|\) is not differentiable at a point
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The function \(|x-a|\) is not differentiable at \(x=a\). Similarly, \(|x+2|\) is not differentiable at \(x=-2\).
Updated On: Jul 3, 2026
- \(x=2\)
- \(x=-2\)
- \(x=-1\)
- \(x=1\)
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The Correct Option is B
Solution and Explanation
Concept:
The modulus function \[ |u| \] is generally not differentiable at the point where \[ u=0 \] Here, \[ u=x+2 \]
Step 1: Find the point where modulus changes sign. \[ x+2=0 \] \[ x=-2 \] So the graph of \[ |x+2| \] has a sharp corner at \[ x=-2 \]
Step 2: Verify using piecewise form. If \[ x+2\geq 0 \] then \[ x\geq -2 \] and \[ |x+2|=x+2 \] If \[ x<-2 \] then \[ |x+2|=-(x+2) \] So, \[ f(x)= \begin{cases} -(x+2), & x\\ x+2, & x\geq -2. \end{cases} \]
Step 3: Compare derivatives. For \(x<-2\), \[ f'(x)=-1 \] For \(x>-2\), \[ f'(x)=1 \] The left derivative and right derivative at \(x=-2\) are not equal. Therefore, the function is not differentiable at \(x=-2\).
Step 4: Final answer. \[ \boxed{x=-2} \]
The modulus function \[ |u| \] is generally not differentiable at the point where \[ u=0 \] Here, \[ u=x+2 \]
Step 1: Find the point where modulus changes sign. \[ x+2=0 \] \[ x=-2 \] So the graph of \[ |x+2| \] has a sharp corner at \[ x=-2 \]
Step 2: Verify using piecewise form. If \[ x+2\geq 0 \] then \[ x\geq -2 \] and \[ |x+2|=x+2 \] If \[ x<-2 \] then \[ |x+2|=-(x+2) \] So, \[ f(x)= \begin{cases} -(x+2), & x\\ x+2, & x\geq -2. \end{cases} \]
Step 3: Compare derivatives. For \(x<-2\), \[ f'(x)=-1 \] For \(x>-2\), \[ f'(x)=1 \] The left derivative and right derivative at \(x=-2\) are not equal. Therefore, the function is not differentiable at \(x=-2\).
Step 4: Final answer. \[ \boxed{x=-2} \]
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