Given the function: find the value of b, if any, that will make the function differentiable at x = 0. a. 0 b. 1 c. 2 d. no such value exists e. there is not enough information to find the value can someone confirm that the answer is d, no such value exists? i don't think there's a value where the function could be continuous and have the same slope on each piece.

You are correct. The answer is choice D

The only way for g(x) to be differentiable at x = 0 is for two things to happen
(1) g(x) is continuous at x = 0
(2) g ' (x) is continuous at x = 0

To satisfy property (1) above, the value of b must be 1. This can be found by plugging x = 0 into each piece of the piecewise function and solving for b. 

So the piecewise function becomes

after plugging in b = 1

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Now differentiate each piece with respect to x to get

The first piece of g ' (x) is always going to be equal to 1. The second piece is equal to zero when x = 0
Because -sin(x) = -sin(0) = 0

So there's this disconnect on g ' (x) meaning its not continuous

Therefore, the value b = 1 will not work. 

So there are no values of b that work to satisfy property (1) and property (2) mentioned at the top.