For what value of the constant c is the function f continuous on (-∞, ∞)?

To determine the value of the constant c for which the function f is continuous on the entire real line, we first need to analyze the function piecewise:

• For x < 2: f(x) = cx² + 2x + 1
• For x > 2: f(x) = cx

For f to be continuous at x = 2, the limits from the left and right must equal f(2). Thus, we first calculate f(2):

Using the left-hand definition:
f(2) = c(2)² + 2(2) + 1 = 4c + 4 + 1 = 4c + 5

Using the right-hand definition:
f(2) = c(2) = 2c

Setting both expressions for f(2) equal gives us:

4c + 5 = 2c

Simplifying this equation:

4c – 2c + 5 = 0

2c + 5 = 0

2c = -5

c = -rac{5}{2}

Thus, for the function f to be continuous on (-∞, ∞), the value of the constant c must be:

-2.5