Let $\mathrm{f}$ be the following piecewise-defined function. \[ f(x)=\left\{\begin{array}{ll} x^{2}+2 & \text { for } x \leq 3 \\ 3 x+2 & \text { for } x>3 \end{array}\right. \] (a) Is $f$ continuous at $x=3$ ? (b) Is $f$ differentiable at $x=3$ ?

Step 1 ：Define the function \(f(x)\) as follows: \[f(x)=\left\{\begin{array}{ll} x^{2}+2 & \text { for } x \leq 3 \\ 3 x+2 & \text { for } x>3 \end{array}\right.\]

Step 2 ：Check if the function is continuous at \(x=3\) by comparing the limit of the function as \(x\) approaches \(3\) from the left, the limit of the function as \(x\) approaches \(3\) from the right, and the value of the function at \(x=3\).

Step 3 ：Calculate the limit of the function as \(x\) approaches \(3\) from the left: \(\lim_{x \to 3^-} f(x) = 11\).

Step 4 ：Calculate the limit of the function as \(x\) approaches \(3\) from the right: \(\lim_{x \to 3^+} f(x) = 11\).

Step 5 ：Calculate the value of the function at \(x=3\): \(f(3) = 11\).

Step 6 ：Since \(\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) = 11\), the function is continuous at \(x=3\).

Step 7 ：Check if the function is differentiable at \(x=3\) by comparing the derivative of the function as \(x\) approaches \(3\) from the left and the derivative of the function as \(x\) approaches \(3\) from the right.

Step 8 ：Calculate the derivative of the function as \(x\) approaches \(3\) from the left: \(f'(3^-) = 6\).

Step 9 ：Calculate the derivative of the function as \(x\) approaches \(3\) from the right: \(f'(3^+) = 3\).

Step 10 ：Since \(f'(3^-) \neq f'(3^+)\), the function is not differentiable at \(x=3\).

Step 11 ：Final Answer: \[\boxed{\text{(a) The function } f \text{ is continuous at } x=3.}\] \[\boxed{\text{(b) The function } f \text{ is not differentiable at } x=3.}\]