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

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Answer 1 · 2017-04-22T02:11:26

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

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

Graphing the function will help to answer the question:

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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 $f(x)$ is continuous at $x=+-1$ the first and second derivatives are not defined at those points.

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:

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

How do you use the first and second derivatives to sketch f(x) = | (x^2) -1 |f(x)=∣∣(x2)−1∣∣f(x) = | (x^2) -1 |?