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

question 398

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.
-
$$y = \frac { 1 } { x } \quad y = - 1$$

• A) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = - 1\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0\), \(f ( x )\) is not differentiable at \(x = - 1\).
• B) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )\) is not differentiable at \(x = - 1\).
• C) Since \(\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )\) is not differentiable at \(x = - 1\).
• D) Since \(\lim _ { x \rightarrow 1 ^ { + } } + f ^ { \prime } ( x ) = 0\) while \(\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0 , f ( x )\) is differentiable at \(x = - 1\).

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