Question

For the following function, find the values of the constants 𝑎 and 𝑏 for which the function is continuous, but not differentiable.

𝑓(𝑥)={𝑎𝑥+𝑏,𝑥>0 & sin𝑥,𝑥≤0.

Answer 1

as with the previous problem, you want

f(0) to match up,
but f'(0) has a break.

how about a=0, b=0
or any linear function f(x) = kx where k≠1
since f'(sinx) = 1 at x=0

Answer 2

[AI-generated answer — Explain Bot]

To find the values of the constants 𝑎 and 𝑏 for which the function 𝑓(𝑥) is continuous but not differentiable, we need to consider the conditions for continuity and differentiability separately.

For a function to be continuous, it needs to be defined at a point and must not have any gaps, jumps, or holes. In this case, 𝑓(𝑥) is defined differently for 𝑥 > 0 and 𝑥 ≤ 0.

- For 𝑥 > 0, 𝑓(𝑥) = 𝑎𝑥 + 𝑏. To ensure continuity, we need the left-hand limit to be equal to the right-hand limit when 𝑥 approaches 0 from the left. This means we need to evaluate 𝑎𝑥 + 𝑏 as 𝑥 approaches 0 from the left. Setting 𝑥 = 0 in the expression 𝑎𝑥 + 𝑏 gives us 𝑏. Therefore, 𝑏 needs to be equal to the value of the function at 𝑥 = 0, which is 𝑓(0) = sin(0) = 0. Hence, 𝑏 = 0.

- For 𝑥 ≤ 0, 𝑓(𝑥) = sin𝑥. Since sin𝑥 is a well-defined function for all values of 𝑥, we don't need to consider any conditions for 𝑎 or 𝑏.

Therefore, the values of 𝑎 and 𝑏 for which the function 𝑓(𝑥) = 𝑎𝑥 + 𝑏 is continuous are 𝑎 = any value and 𝑏 = 0.

Now let's discuss the differentiability of 𝑓(𝑥). For a function to be differentiable at a point, it needs to have a well-defined derivative at that point, which means that the derivative from the left-hand side should be equal to the derivative from the right-hand side.

Looking at 𝑓(𝑥), we observe that the derivative of 𝑓(𝑥) = 𝑎𝑥 + 𝑏 with respect to 𝑥 is 𝑎. This derivative is a constant and does not depend on 𝑥. On the other hand, when 𝑥 ≤ 0, the derivative of 𝑓(𝑥) = sin𝑥 with respect to 𝑥 is cos𝑥, which is defined for all 𝑥.

Since the derivative of 𝑓(𝑥) = 𝑎𝑥 + 𝑏 is a constant (𝑎) and the derivative of 𝑓(𝑥) = sin𝑥 is defined for all values of 𝑥, the function 𝑓(𝑥) is differentiable for any values of 𝑎 and 𝑏.

Therefore, the values of 𝑎 and 𝑏 for which the function 𝑓(𝑥) is continuous but not differentiable are 𝑎 = any value and 𝑏 = 0.