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) :} #