A) Continuity at $x = 1$

For the function $f(x)$ to be continuous at $x = 1$, the left-hand limit (LHL) and right-hand limit (RHL) at $x = 1$ must be equal to the value of the function at $x = 1$.

1. Left-hand limit (LHL) as $x \to 1^-$:

The piece of the function when $x < 1$ is given by: $f(x) = 3 - x$

As $x \to 1^-$: $\lim_{x \to 1^-} f(x) = 3 - 1 = 2$

2. Right-hand limit (RHL) as $x \to 1^+$:

The piece of the function when $x \geq 1$ is given by: $f(x) = ax^2 + bx$

As $x \to 1^+$: $\lim_{x \to 1^+} f(x) = a(1)^2 + b(1) = a + b$

For continuity, the limits must match: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x)$ $2 = a + b$

Thus, the relationship between $a$ and $b$ is: $a + b = 2$

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B) Continuity and Differentiability at $x = 1$

For $f(x)$ to be differentiable at $x = 1$, the derivatives from the left and right must also be equal.

1. Left-hand derivative (LHD) as $x \to 1^-$:

The derivative of $f(x) = 3 - x$ when $x < 1$ is: $f'(x) = -1$

Thus: $\lim_{x \to 1^-} f'(x) = -1$

2. Right-hand derivative (RHD) as $x \to 1^+$:

The derivative of $f(x) = ax^2 + bx$ when $x \geq 1$ is: $f'(x) = 2ax + b$

As $x \to 1^+$: $\lim_{x \to 1^+} f'(x) = 2a(1) + b = 2a + b$

For differentiability, the left and right derivatives must match: $\lim_{x \to 1^-} f'(x) = \lim_{x \to 1^+} f'(x)$ $-1 = 2a + b$

3. System of Equations:

From continuity: $a + b = 2$

From differentiability: $2a + b = -1$

Subtract the first equation from the second: $(2a + b) - (a + b) = -1 - 2$ $a = -3$

Substitute $a = -3$ into $a + b = 2$: $-3 + b = 2$ $b = 5$

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Final Answer:

The unique values of $a$ and $b$ that make $f(x)$ both continuous and differentiable at $x = 1$ are: $a = -3, \quad b = 5$

Verification:

1. Continuity: $a + b = -3 + 5 = 2 \quad \text{(satisfied)}$

2. Differentiability: $2a + b = 2(-3) + 5 = -6 + 5 = -1 \quad \text{(satisfied)}$

Would you like further clarification on any step or additional examples?

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Follow-Up Questions:

1. Why does continuity imply matching function values at $x = 1$?
2. How does differentiability rely on the function's derivative?
3. What are the implications of discontinuity at $x = 1$?
4. How does this process change if $f(x)$ were a piecewise-defined trigonometric function?
5. What role does the smoothness of polynomials play in differentiability?

Tip:

Always check both continuity and differentiability when working with piecewise functions at critical points.