I think the correct answer is the ONLY first one. Am I right?

Which of the following properties are satisfied by the function
f(x)=

x^2+1∣x<0

1∣x=0

5x+1∣x>0

(I) f(x) is continuous
(II) f(x) is differentiable for all x
(III) f(x) is differentiable at x = -2

1 Answer
Jun 7, 2018

The correct answers are the first and the third.

Explanation:

First Answer
The function is indeed continuous. In fact, it is composed by three continuous patched, which connect continously at x=0, since

\lim_{x \to 0^-} x^2+1 = \lim_{x \to 0^+} 5x+1 = f(0)

Second Answer
The function is not differentiable everywhere: the derivative for negative x values is 2x, and for positive x values is 5. These two function do not meet continuously at x=0

Third answer
The function is differentiable at x=-2. In fact, in this case we use the function x^2+1, which is differentiable at x=-2.