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Problem 74 Hard Difficulty

Where is the function $ h(x) = \mid x - 1 \mid + \mid x + 2 \mid $ differentiable? Give a formula for $ h' $ and sketch the graph of $ h $ and $ h'. $

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Video Transcript

Hey, it's clear. So when you read here so we have Agent X is equal to the absolute value of X minus one was the absolute value of X plus two is equal to negative X minus one minus X plus two. This is for X is less than or equal to negative, too. We have negative X minus one less X plus two, and this is for access between negative two and one. We have X minus one less X plus two. This is for X is larger than one. These equations can be rewritten and simplified. Negative two X minus one. This can be three, and this is two x plus one When we differentiate it, we had the following negative, too access less than or equal to negative, too. Zero for access between negative two and one and two for X is bigger than one. So we see that it's not continuous, but negative two and one so it does not exist. That X is equal to negative two and one. When we take the formula, we get the absolute value of X plus two over X plus two. What's the absolute value of X minus one what were X minus one? We're going to also draw a graph where reference When we get the original graph. Just a Jiff. X looks like this. And then we have our derivative grab in red.