# How do you use the first and second derivatives to sketch #f(x) = | (x^2) -1 |#?

##### 1 Answer

Apr 22, 2017

# f'(x) \ = { (2x-1,,x lt -1), (1-2x,, -1 lt x lt 1), (2x-1,, x gt 1) :} #

# f''(x) = { (2,,x lt -1), (-2,, -1 lt x lt 1), (2,, x gt 1) :} #

#### Explanation:

Graphing the function will help to answer the question:

So we can write the function as:

# f(x) = |x^2-1 | #

# " " = { (x^2-1,,x^2-1 gt 0), (-(x^2-1),,x^2-1 lt 0) :} #

# " " = { (x^2-1,,x lt -1), (1-x^2,, -1 lt x lt 1), (x^2-1,, x gt 1) :} #

Note that although

So then we can easily differentiate to get the first derivative:

# f'(x) \ = { (2x-1,,x lt -1), (1-2x,, -1 lt x lt 1), (2x-1,, x gt 1) :} #

And the second derivative is:

# f''(x) = { (2,,x lt -1), (-2,, -1 lt x lt 1), (2,, x gt 1) :} #