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

question 136

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

---

Definitions:

Quantity Supplied

The total amount of a good or service that producers are willing and able to sell at a given price level and period.

Price

The sum projected, mandated, or allocated in money for acquiring something.

Perfectly Price Discriminate

A theoretical pricing strategy where a seller charges the maximum possible price for each unit sold, capturing the entire consumer surplus.

Monopolist

A market player that is the sole provider of a particular good or service, possessing significant market power to influence prices and output levels.