Compare the Right-Hand and Left-Hand Derivatives to Determine Whether or Not

question 168

Multiple Choice

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
- $y=x^{2} \quad y=\sqrt{x}$

• A) Since \(\lim _ { x \rightarrow1 ^ { + } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2\) while \(\lim _ { x \rightarrow 1 ^ { - } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 1 } { 2 } , \mathrm { f } ( \mathrm { x } )\) is not differentiable at \(x = 1\).
• B) Since \(\lim _ { x \rightarrow 1^ { + } } \mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { 2 }\) while \(\lim _ { x \rightarrow 1 ^ { - } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = 1 , \mathrm { f } ( \mathrm { x } )\) is not differentiable at \(x = 1\).
• C) Since \(\lim _ { x \rightarrow1 ^ { + } } f ^ { \prime } ( x ) = \frac { 1 } { 2 }\) while \(\lim _ { x \rightarrow 1 ^ { - } }f ^ { \prime } ( x ) = 2 , f ( x )\) is not differentiable at \(x = 1\).
• D) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 2\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )\) is differentiable at \(x = 1\).

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