To determine which function is differentiable at $x = 3$, we need to check two conditions:

1. Continuity at $x = 3$: The left-hand limit ($\lim_{x \to 3^-} f(x)$) and right-hand limit ($\lim_{x \to 3^+} f(x)$) must match and equal $f(3)$.
2. Derivative matching at $x = 3$: The left-hand derivative ($\lim_{x \to 3^-} f'(x)$) and right-hand derivative ($\lim_{x \to 3^+} f'(x)$) must be equal.

Analyze Each Option

(A) $f(x) = \begin{cases} x^2 + 1, & x \leq 3 \\ x^2, & x > 3 \end{cases}$

• Continuity: $\text{For } x \leq 3, \quad f(x) = x^2 + 1 \implies f(3) = 3^2 + 1 = 10.$ $\text{For } x > 3, \quad f(x) = x^2 \implies \lim_{x \to 3^+} f(x) = 3^2 = 9.$ Since $\lim_{x \to 3^-} f(x) = 10$ and $\lim_{x \to 3^+} f(x) = 9$, the function is not continuous. Hence, not differentiable.

(B) $f(x) = \begin{cases} x^2 + 1, & x \leq 3 \\ 6x - 8, & x > 3 \end{cases}$

• Continuity: $\text{For } x \leq 3, \quad f(3) = 3^2 + 1 = 10.$ $\text{For } x > 3, \quad f(x) = 6x - 8 \implies \lim_{x \to 3^+} f(x) = 6(3) - 8 = 10.$ Since $\lim_{x \to 3^-} f(x) = 10$ and $\lim_{x \to 3^+} f(x) = 10$, the function is continuous.
• Derivative: $\text{Left-hand derivative: } f'(x) = 2x \text{ for } x \leq 3 \implies f'(3^-) = 2(3) = 6.$ $\text{Right-hand derivative: } f'(x) = 6 \text{ for } x > 3 \implies f'(3^+) = 6.$ Since $f'(3^-) = f'(3^+) = 6$, the function is differentiable.

(C) $f(x) = \begin{cases} x^2 + 1, & x \leq 3 \\ 6, & x > 3 \end{cases}$

• Continuity: $\text{For } x \leq 3, \quad f(3) = 3^2 + 1 = 10.$ $\text{For } x > 3, \quad f(x) = 6 \implies \lim_{x \to 3^+} f(x) = 6.$ Since $\lim_{x \to 3^-} f(x) = 10 \neq 6$, the function is not continuous, so it is not differentiable.

(D) $f(x) = \begin{cases} x^2 + 1, & x \leq 3 \\ 10, & x > 3 \end{cases}$

• Continuity: $\text{For } x \leq 3, \quad f(3) = 3^2 + 1 = 10.$ $\text{For } x > 3, \quad f(x) = 10 \implies \lim_{x \to 3^+} f(x) = 10.$ Since $\lim_{x \to 3^-} f(x) = 10$ and $\lim_{x \to 3^+} f(x) = 10$, the function is continuous.
• Derivative: $\text{Left-hand derivative: } f'(x) = 2x \text{ for } x \leq 3 \implies f'(3^-) = 2(3) = 6.$ $\text{Right-hand derivative: } f'(x) = 0 \text{ for } x > 3 \implies f'(3^+) = 0.$ Since $f'(3^-) \neq f'(3^+)$, the function is not differentiable.

Conclusion:

The only function differentiable at $x = 3$ is (B).

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Related Questions:

1. What is the formal definition of differentiability at a point?
2. How can we verify continuity of piecewise functions more systematically?
3. Why is differentiability dependent on both continuity and matching derivatives?
4. Can a function be continuous but not differentiable? Provide an example.
5. How does the choice of piecewise conditions affect differentiability?

Tip: Always check continuity first when determining differenti