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Let a function g : [0, 4] → R be defined as g(x) = { max_{0≤t≤x} (t^3 − 6t^2 + 9t − 3), 0 ≤ x ≤ 3 { 4 − x, 3 < x ≤ 4 then the number of points in the interval (0, 4) where g(x) is NOT differentiable, is _____.

1. 1
2. 2
3. 3
4. 4

Correct answer: 4

Solution

The function g(x) is defined piecewise, with a change in definition at x = 3. The first part involves a cubic function which can have critical points where it is not differentiable, and the second part is a linear function that meets the first part at x = 3. The transition at x = 3 and potential critical points from the cubic function contribute to a total of 4 points where g(x) is not differentiable.

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