At how many points is f(x) = | (x - 1)^2(x - 3) | not differentiable?

Question 125

HardOriginal question

At how many points is $f(x) = \lvert (x - 1)^{2}(x - 3) \rvert$ not differentiable?

0

Correct answerYour answer

1

Correct answerYour answer

2

Correct answerYour answer

3

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$u = (x - 1)^{2}(x - 3)$ is 0 at $x = 1$ and $x = 3$ . At 3 it changes sign, so $\lvert u \rvert$ has a corner there, with slopes $-4$ and 4.
At 1 the double root keeps $u \le 0$ on both sides, so $f = -u$ near 1, a smooth curve with slope 0. So 1 point.

Worth noting

2 counts the double root 1 as well; there $u$ touches 0 without changing sign, so the modulus does not fold it.

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