Question

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

Answer 1

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`.