SOLUTION: Let f(x)={&#8722;6x^2+2x for x<0, f(x)={3x^2-2 for x&#10878;0. According to the definition of the derivative, to compute f&#8242;(0), we need to compute the left-hand limi

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Question 1075732: Let
f(x)={−6x^2+2x for x<0,
f(x)={3x^2-2 for x⩾0.
According to the definition of the derivative, to compute f′(0), we need to compute the left-hand limit
limx→0− =?
, which is
undefined
,
and the right-hand limit
limx→0+?
, which is
0
.
We conclude that f′(0) is
undefined
.
Note: If a limit or derivative is undefined, enter 'undefined' as your answer.
Found 2 solutions by Boreal, ikleyn:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
The right hand derivative is 6x, and at 0 the derivative is 0. Furthermore, as x approaches 0 from the right, the limit approaches 0. Answer for the right-hand limit is 0.
For the left hand limit, the derivative is -12x+2, and the limit is +2. As the function approaches 0 from the left, it doesn't exist at 0, but it does exist for points less than 0, and the value approaches +2.
Because +2 and 0 are not equal, f'(0) is undefined.
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
Let
f(x)={−6x^2+2x  for x<0,
f(x)={3x^2-2     for x⩾0.

According to the definition of the derivative, to compute f′(0), we need to compute the left-hand limit 

limx→0− =?

 , which is 
undefined                   <<<---+++ W R O N G
 , 

and the right-hand limit 

limx→0+?

 , which is 
0                           <<<---+++ C O R R E C T
 . 

We conclude that f′(0) is 
undefined
 .


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