Find the values of 𝑎 and 𝑏 so that the following function is differentiable at 𝑥 = 0. 𝑓 (𝑥) = { 𝑥 + 𝑎 if 𝑥 < 0 𝑒 𝑏𝑥 − 1 if 𝑥 ≥ 0

For the function to be differentiable at 𝑥 = 0, it needs to have a defined derivative at that point. This means that the left-hand and right-hand limits of the function at 𝑥 = 0 must be equal, and the derivative at 𝑥 = 0 must exist.

To find the limit from the left, we can substitute a small negative number, say -𝛿, for 𝑥 in the expression 𝑥 + 𝑎:

lim (𝑥→0-) 𝑓 (𝑥) = lim (𝑥→0-) 𝑥 + 𝑎 = 𝑎

Similarly, to find the limit from the right, we can substitute a small positive number, say 𝛿, for 𝑥 in the expression 𝑒 𝑏𝑥 − 1:

lim (𝑥→0+) 𝑓 (𝑥) = lim (𝑥→0+) 𝑒 𝑏𝑥 − 1 = 𝑏

For the function to be differentiable at 𝑥 = 0, we need 𝑎 = 𝑏. This is because the derivative of the function at 𝑥 = 0 is given by the limit:

f'(0) = lim (𝑥→0) [f(𝑥) - f(0)] / 𝑥

Substituting the values of 𝑎 and 𝑏, we get:

f'(0) = lim (𝑥→0) [(𝑥 + 𝑏) - 𝑏] / 𝑥 = 1

Therefore, 𝑎 = 𝑏 = 1 for the function to be differentiable at 𝑥 = 0.