Find all the points where the fumction is not differentiable: f(x)=∥x∥
The Correct Answer and Explanation is:
The function ( f(x) = |x| ) refers to the absolute value of ( x ), where ( |x| = |x| ). This is a piecewise function defined as:
[
f(x) =
\begin{cases}
x, & \text{if } x \geq 0, \
-x, & \text{if } x < 0.
\end{cases}
]
To find the points where the function is not differentiable, we need to consider where the function might have sharp corners or discontinuities, as these are typical places where differentiability fails.
Step 1: Check the function at ( x = 0 )
The function ( f(x) = |x| ) has a sharp corner at ( x = 0 ). At this point, the left-hand and right-hand derivatives are not equal:
- Right-hand derivative: For ( x > 0 ), the function ( f(x) = x ). The derivative of ( f(x) ) is ( f'(x) = 1 ) for ( x > 0 ).
- Left-hand derivative: For ( x < 0 ), the function ( f(x) = -x ). The derivative of ( f(x) ) is ( f'(x) = -1 ) for ( x < 0 ).
At ( x = 0 ), the right-hand derivative is ( 1 ) and the left-hand derivative is ( -1 ). Since the left-hand and right-hand derivatives do not match, the function is not differentiable at ( x = 0 ).
Step 2: Check the function for other points
For any ( x \neq 0 ), the function ( f(x) = |x| ) behaves as either ( f(x) = x ) or ( f(x) = -x ), both of which are differentiable. Specifically:
- For ( x > 0 ), ( f(x) = x ) is differentiable with derivative ( f'(x) = 1 ).
- For ( x < 0 ), ( f(x) = -x ) is differentiable with derivative ( f'(x) = -1 ).
Conclusion:
The function ( f(x) = |x| ) is differentiable for all ( x \neq 0 ), but it is not differentiable at ( x = 0 ) due to the discontinuity in the derivative at that point. Therefore, the function is not differentiable only at ( x = 0 ).