The function `g(x) = {{:(x+b",",x lt 0),(cos x",",x ge 0):}` can be made differentiable at x = 0

Options

A. (a) if b is equal to zero
B. (b) if b is not equal to zero
C. (c) if b takes any real value
D. (d) for no value of b

Answer

To determine the value of \( b \) for which the function

\[ g(x) = \begin{cases} x + b & \text{if } x < 0 \\ \cos x & \text{if } x \geq 0 \end{cases} \]

can be made differentiable at \( x = 0 \), we need to ensure that both the function is continuous at \( x = 0 \) and that the left-hand derivative equals the right-hand derivative at that point.

### Step 1: Check Continuity at \( x = 0 \)

For \( g(x) \) to be continuous at \( x = 0 \), we need:

\[ \lim_{x \to 0^-} g(x) = \lim_{x \to 0^+} g(x) = g(0) \]

Calculating \( g(0) \):

\[ g(0) = \cos(0) = 1 \]

Now, calculate the left-hand limit:

\[ \lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} (x + b) = 0 + b = b \]

Now, calculate the right-hand limit:

\[ \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} \cos x = \cos(0) = 1 \]

Setting the left-hand limit equal to the right-hand limit for continuity:

\[ b = 1 \]

### Step 2: Check Differentiability at \( x = 0 \)

Next, we need to check the derivatives from both sides.

**Left-hand derivative:**

\[ g'(x) = 1 \quad \text{for } x < 0 \]

Thus, the left-hand derivative at \( x = 0 \) is:

\[ \lim_{x \to 0^-} g'(x) = 1 \]

**Right-hand derivative:**

\[ g'(x) = -\sin x \quad \text{for } x \geq 0 \]

Thus, the right-hand derivative at \( x = 0 \) is:

\[ \lim_{x \to 0^+} g'(x) = -\sin(0) = 0 \]

### Step 3: Compare the Left-hand and Right-hand Derivatives

For \( g(x) \) to be differentiable at \( x = 0 \):

\[ \lim_{x \to 0^-} g'(x) = \lim_{x \to 0^+} g'(x) \]

This means:

\[ 1 \neq 0 \]

Since the left-hand derivative (1) does not equal the right-hand derivative (0), the function is not differentiable at \( x = 0 \) for any value of \( b \).

### Conclusion

Thus, the function \( g(x) \) can be made differentiable at \( x = 0 \) for **no value of \( b \)**.

### Final Answer

The answer is: **For no value of \( b \)**.

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