Final answer:
To make the function continuous, we equate the function pieces and solve for c and d.
Step-by-step explanation:
To find the values of c and d that make the function continuous for all x, we need to ensure that the function pieces fit together at their boundaries. In this case, the boundaries are x = 1 and x = 2.
When x < 1, f(x) = 9x.
When 1 ≤ x < 2, f(x) = cx² + d.
When x ≥ 2, f(x) = 3x.
We need to find the values of c and d such that f(x) is continuous when x = 1 and when x = 2. To do this, we equate the two function pieces at their boundaries.
At x = 1: 9(1) = c(1)² + d.
At x = 2: 3(2) = c(2)² + d.
We can solve these two equations to find the values of c and d.