Let f be the function defined above. Which of the following statements is true?

Question

f (x)= { 2 for x<5

{ 2x-4 for x≥5

Let f be the function defined above. Which of the following statements is true?

A) f is not differentiable at x = 5 because f is not continuous at x = 5 .

B) f is not differentiable at x = 5 because the graph of f has a sharp corner at x = 5 .

C) f is not differentiable at x = 5 because the graph of f has a vertical tangent at x = 5

D) f is not differentiable at x = 5 because f is not defined at x = 5 .

Answer 1

The answer is A. If f(x) is evaluated at x = 5, the part of f(x) in the interval x < 5 yields 2 while the part in the interval x >= 5 yields a value of 6. The difference in value allows us to conclude that f(x) is not continuous at x = 5.

You could also use the limit of f(x) as x approaches 5 from the negative and positive side. You will discover that as x approaches 5 from the negative side, f(x) has a value of 2. As x approaches 5 from the positive side, f(x) has a value of 6. Since the values of f(x) are not equal the function is not continuous.