Consider the following two statements about the function f(x) = |x|: P. f(x)…
2007
Consider the following two statements about the function f(x) = |x|:
P. f(x) is continuous for all real values of x.
Q. f(x) is differentiable for all real values of x.
Which of the following is TRUE?
- A.
P is true and Q is false.
- B.
P is false and Q is true.
- C.
Both P and Q are true.
- D.
Both P and Q are false.
Show answer & explanation
Correct answer: A
For f(x) = |x|:
Continuity: |x| is continuous for every real value of x, including x = 0. Hence statement P is true.
Differentiability: For x > 0, f(x) = x, so the derivative is 1. For x < 0, f(x) = -x, so the derivative is -1. At x = 0, the left-hand derivative is -1 and the right-hand derivative is 1.
Since these one-sided derivatives are not equal, f(x) is not differentiable at x = 0. Hence statement Q is false.
Therefore, P is true and Q is false.