Question

Given the function

g(x)={x+b, x<0
cos(x),​x≥0​.
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 determine the value.

Answer 1

To make the function g(x) differentiable at x=0, b must equal 1 for continuity and the derivatives must match from both sides at x=0. Since b=1 provides the required continuity and the derivatives match, option (B) is the correct answer.

Step-by-step explanation:

To determine the value of b that makes the function g(x) differentiable at x=0, we need to ensure continuity and the equality of derivatives from both sides of x=0. For a piecewise function like g(x), which has g(x)=x+b for x<0 and g(x)=cos(x) for x≥0, continuity implies that the limit of g(x) as x approaches 0 from the left should equal the limit as x approaches 0 from the right.

For continuity at x=0:

• lim x0 (x+b) = 0 + b = b

• lim x0 cos(x) = cos(0) = 1

Hence, b must equal 1 for the limits to match. For differentiability at x=0, the derivative from the left should equal the derivative from the right:

• Derivative from the left: 1

• Derivative from the right: -sin(0) = 0

Since the derivative of cos(x) is -sin(x) and sin(0) is 0, the derivatives match. Therefore, the function is differentiable at x=0 when b=1, which corresponds to option B in the final answer.