f(x) = 3x+5 if x<0, f(x) = sin(6ax)+b if x>0, if f(x) is continuous and f’(0) exists, find a and b

To ensure that \( f(x) \) is continuous and differentiable at \( x = 0 \), we must make sure that both \( f(x) \) and \( f'(x) \) match at \( x = 0 \) from the left and the right.

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

For \( f(x) \) to be continuous at \( x = 0 \):
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0).
\]

**Left-hand limit** (\( x \to 0^- \)):
Since \( f(x) = 3x + 5 \) when \( x < 0 \),
\[
\lim_{x \to 0^-} f(x) = 3(0) + 5 = 5.
\]

**Right-hand limit** (\( x \to 0^+ \)):
Since \( f(x) = \sin(6ax) + b \) when \( x > 0 \),
\[
\lim_{x \to 0^+} f(x) = \sin(0) + b = b.
\]

For continuity, these limits must be equal:
\[
5 = b.
\]

Thus, \( b = 5 \).

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

For \( f'(0) \) to exist, the left-hand and right-hand derivatives at \( x = 0 \) must be equal.

**Left-hand derivative** (\( x < 0 \)):
\[
f(x) = 3x + 5 \Rightarrow f'(x) = 3.
\]
So,
\[
\lim_{x \to 0^-} f'(x) = 3.
\]

**Right-hand derivative** (\( x > 0 \)):
Since \( f(x) = \sin(6ax) + 5 \), we differentiate \( f(x) \) with respect to \( x \):
\[
f'(x) = 6a \cos(6ax).
\]
Thus,
\[
\lim_{x \to 0^+} f'(x) = 6a \cos(0) = 6a.
\]

For \( f'(0) \) to exist, we must have:
\[
3 = 6a.
\]
Solving for \( a \):
\[
a = \frac{1}{2}.
\]

### Conclusion

The values of \( a \) and \( b \) that make \( f(x) \) continuous and differentiable at \( x = 0 \) are:
\[
a = \frac{1}{2}, \quad b = 5.
\]

Would you like a more detailed explanation, or do you have any questions?

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**Related Questions:**

1. What is the role of continuity and differentiability in defining a function's smoothness?
2. How would the solution change if \( f(x) \) had a different expression for \( x < 0 \)?
3. What happens if only continuity is required and not differentiability?
4. How does differentiability at a point imply continuity, but not vice versa?
5. Could there be any other values for \( a \) and \( b \) if different conditions were given?

**Tip:** For functions with piecewise definitions, always start by checking continuity and then differentiability at the boundaries.

Explore the continuity and differentiability of a piecewise-defined function, f(x) = 3x+5 for x<0 and f(x) = sin(6ax)+b for x>0. Learn how to find constants a and b to ensure f(x) is smooth and differentiable at x=0. This solution covers limit-based continuity and differentiability conditions.

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