Question

Or what value of the constant c is the function f continuous on (−[infinity], [infinity])?

f(x) =
A.cx2 + 2x
B.x3 − cx C.if x < 5 D.if x ≥ 5

Answer 1

To make the function continuous, we need to ensure that both parts of the function are continuous at x = 5. By taking limits as x approaches 5 on both sides of the function and equating them, we can find the values of A and B, and then solve for c.

Step-by-step explanation:

To determine the value of the constant c that makes the function f continuous on the interval (-∞, ∞), we need to ensure that both parts of the function, A.cx² + 2x and B.x³ - cx, are continuous at the point where their domains meet, which is x = 5.

For the left side of the function, which is A.cx² + 2x, we pick a value larger than 5, say x = 6. We set up a limit as x approaches 5 to find the value of A. So, taking the limit as x approaches 5, we have:
limx→5 (A.cx² + 2x) = A(5)² + 2(5)

For the right side of the function, which is B.x³ - cx, we pick a value smaller than 5, say x = 4. We set up a limit as x approaches 5 to find the value of B. So, taking the limit as x approaches 5, we have:
limx→5 (B.x³ - cx) = B(5)³ - c(5)

Now, equate the two limits and solve for A and B to find the value of c that makes the function continuous.