Final answer:
The piecewise function is continuous at x=3 because both the left-hand and right-hand limits match, and the function equals 9. It is differentiable because the derivatives from both sides are equal when x=3. Thus, the function is both continuous and differentiable at x=3.
Step-by-step explanation:
At x=3, we have a piecewise function defined by f(x) = x² for x<3 and f(x) = 6x-9 for x≥3. To determine the behavior of the function at x=3, we first need to evaluate the left-hand limit (as x approaches 3 from the left) and the right-hand limit (as x approaches 3 from the right), as well as the value of the function at x=3.
If x<3, f(x) is defined as x². Therefore, the value of f(x) as x approaches 3 from the left is 3² or 9. If x≥3, f(x) is defined as 6x-9. Therefore, the value of f(x) as x approaches 3 from the right is 6(3)-9, which is also 9. This means the function is continuous at x=3 because the left-hand and right-hand limits are equal, and the value of the function at x=3 also equals 9.
To check if the function is differentiable, we look at the derivative of the function on either side of x=3. For x<3, the derivative of f(x) = x² is 2x, which is 6 when x=3. For x≥3, the derivative of f(x) = 6x-9 is 6, which is also consistent when x=3. Since the derivative from the left is equal to the derivative from the right at x=3, the function is also differentiable at that point.
Therefore, the correct option in the final answer is (E) both continuous and differentiable.