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I'm just doing some math and I could use some help. The problem asks me to draw a graph based on the following properties:

$f'(x) > 0$ if $\lvert x\rvert < 2$

$f'(x) < 0$ if $\lvert x\rvert > 2$

$f'(-2) = 0$

$\lim_{x\to2}\lvert f'(x)\rvert = \infty$

$f''(x) > 0$ if $x \neq 2$

What confuses me is the absolute value of the derivative approaches infinity as x approaches two, could someone explain what this means for the original function?

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This could mean that $f'(x)$ tends to $\infty$ on both sides or to $-\infty$ on both sides or one of each. You have to use the information given in the first two properties to determine which case the function falls into. Depending on which case you find in the above tells us whether $f$ increases or decreases in a neighbourhood of $2$.

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