Graphing Piecewise Functions Using Derivatives

[farmd684]

Homework Statement

y=|x^2-4|+2 find f`,relative extrema,minima.

Homework Equations

Detailed graphing of
y={ 9-x,x<=3
x^2-3,x>3

using derivatives

The Attempt at a Solution

I m not sure how to do this sort of prblems.

 

[CompuChip]

Usually, it works to get rid of the absolute value signs. So first solve where x^2 - 4 is positive and negative and treat those cases separately.

Simple example: derive y = |x|.
1) For x > 0 (or x = 0), this reads y = x. Then the derivative is y' = 1.
2) For x < 0, it means y = -x. Then the derivative is y' = -1.

Afterwards, you may try to find a general formula. For example, if you are used to working with such things, you might write something like: y' = sign(x) for x non-zero (and for x = 0, the derivative is undefined).

 

[HallsofIvy]

Graphing is the simplest thing to do. As CompuChip said, get rid of the absolute value signs. x²- 4 is 0 at x= -2 and x= 2. Between -2 and 2, x²- 4 is negative and so |x²- 4|= 4- x². Graph y= x²- 4 up to x= -2, then y= 4- x² for x between -2 and 2, and, finally, graph y= x²- 4 for x> 2.

That is the same, of course, as graphing y= x²- 4, then "flipping" the part of the graph below the y-axis up. It should be easy to see where relative max and min are.

 

[CompuChip]

[...]
Graph y= x²- 4 up to x= -2, then y= 4- x² for x between -2 and 2, and, finally, graph y= x²- 4 for x> 2.

That is the same, of course, as graphing y= x²- 4, then "flipping" the part of the graph below the y-axis up. [..]

And for your real question, don't forget the +2

 

[farmd684]

And for your real question, don't forget the +2

Thanks i have understood that.I know how to graph piecewise functions but i m confused how to draw them by using the method of derivative i mean by f` and f`` increasing part decreasing part,concave up down and so on.