Question

Find the values of a and b that make f continuous everywhere. f(x) = x² - 4x - 2 if x < 2, ax² - bx³ if 2 ≤ x < 3, and 2x - ab if x ≥ 3.

Answer 1

To find the values of a and b that make f continuous everywhere, set f(2) equal to f(3) and solve for a and b.

Step-by-step explanation:

To find the values of a and b that make f continuous everywhere, we need to find where the different pieces of f(x) meet at critical points. Since f(x) is continuous at x = 2 and x = 3, these points must be the same for the two pieces of the function. Therefore, we need to calculate the values of a and b that make f(2) equal to f(3).

For x < 2, we have f(x) = x² - 4x - 2.

For 2 ≤ x < 3, we have f(x) = ax² - bx³.

For x ≥ 3, we have f(x) = 2x - ab.

By setting f(2) equal to f(3) and solving for a and b, we can find the values that make f continuous everywhere.