Question

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

f(x)= cx² + 6x if x < 3

x³ − cx if x ≥ 3

Answer 1

For the function to be continuous on (-∞, ∞), the constant c must be such that the function's value and rate of change are the same on both sides of x = 3. Solving this for c gives us c = 1.5.

Step-by-step explanation:

The question asks for the value of the constant c that makes the function f(x) continuous on the interval −∞ to ∞. To be continuous, f(x) must be both unbroken (no gaps) and smooth (no sharp turns or cusps) across its domain. This means that both pieces of the piecewise function must equal each other at the point where they switch, which is at x = 3.

We want the following to be true for continuity at x = 3:

Limit as x approaches 3 from the left of f(x) = Limit as x approaches 3 from the right of f(x) = f(3)

This gives us two expressions based on the given piecewise function:

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• 
  

Equate the two functions at x = 3:

3c + 18 = 27 - 3c

Solving for c, we get:

6c = 9

c = 1.5

Thus, the function f(x) is continuous for the value of c being 1.5.