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Numerade Educator



Problem 73 Hard Difficulty

(a) For what values of $ x $ is the function $ f(x) = \mid x^2 - 9 \mid $ differentiable? Find a formula for $ f'. $
(b) Sketch the graph of $ f $ and $ f'. $


(a) $f$ is not differentiable at 3 or at -3

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Video Transcript

you know, it's clear a seven inning right here. So we have X squared minus nine. The absolute value we're first going to calculate. We're making a equal to zero. Let me get access equal to positive or negative three. So f of X is not different. Shovel there. We know that Y is equal to X square minus nine. This is gonna be an upward open opening parabola aboard opening crapola and have Ciro's a positive and negative three. So when access less than or equal to negative three and when access bigger or equal to three you have f of X is equal to X square minus nine. We don't need the absolute value signs, since it's already gonna be positive. And when access between negative three and three we get half of X is equal to negative X Square minus nine, which gives us negative X Square plus nine. So when we differentiate for the first part, we get the derivative where X Square minus nine is equal to two x and for access between negative three and three. This is negative. X Square plus nine. We get negative to USC's. So we our answer is it is not different. Chewable at X is equal to positive or negative free and are derivative. Is equal to two X If X is less than negative. Three or ex this bigger than three. Negative to X for access between negative three and positive three. Next, we're going to grab this. So if we grab our original well, look like this then are derivative says that six negative six.