For what value of the constant c is the function f continuous on (-∞, ∞)? f(x) = {cx² + 2x/x² - 1 if x < 2 and x³ - cx if x >= 2

Solution:

Given, the function f(x) = {cx²+ 2x/x² - 1 if x < 2 and x³ - cx if x >= 2 is continuous on (-∞, ∞).

We have to find the value of the constant c.

We know that f(x) is continuous at x = a if the following conditions are satisfied.

1) f(a) exists.

2) \(\lim_{x \to a}\) f(x) exists and

3) \(\lim_{x \to a}\) f(x) = f(a).

So, we know that for any value of c, for x<2, the function is continuous and for x>=2, the function is continuous.

Therefore, it is the case that we have to check for the continuity of the function at x = 2.

For any value of c, the first two conditions are satisfied. 

So, to find c, assume that the third condition is satisfied inorder to make the function continuous.

Now, \(\lim_{x \to 2}\) f(x) = f(2).

\(\lim_{x \to 2}\) f(x) = cx² + 2x/x² - 1

f(2) = c(2)² + 2(2)/(2)² - 1

f(2) = 4c - 1

\(\lim_{x \to 2}\) f(x) = x³ - cx

f(2) = (2)³ - c(2)

f(2) = 8 - 2c

So, 4c - 1 = 8 - 2c

4c + 2c = 8 + 1

6c = 9

c = 9/6

c = 3/2

c = 1.5

Therefore, the value of c is 1.5

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For what value of the constant c is the function f continuous on (-∞, ∞)? f(x) = {cx² + 2x/x² - 1 if x < 2 and x³ - cx if x >= 2

Summary:

The function f is continuous on (-∞, ∞), f(x) = {cx² + 2x/x² - 1 if x < 2 and x³ - cx if x >= 2, the value of the constant c is 1.5