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
KillerBunny
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#.

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