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 4x - ab if x ≥ 3.

Answer 1

To make the function f continuous everywhere, we need to ensure that the different pieces of the function match up at the points where they connect. In this case, we have three different pieces. The first step is to find the values of a and b that make the function continuous at x = 2 and x = 3. Solving the equations that result from setting the different pieces of the function equal to each other at these points will give us the values of a and b that make the function continuous.

Step-by-step explanation:

To make the function f continuous everywhere, we need to ensure that the different pieces of the function match up at the points where they connect. In this case, we have three different pieces:

1. For x < 2, f(x) = x² - 4x - 2
2. For 2 ≤ x < 3, f(x) = ax² - bx³
3. For x ≥ 3, f(x) = 4x - ab

The first step is to find the values of a and b that make the function continuous at x = 2 and x = 3. To do this, we set the first piece and the second piece equal to each other at x = 2, and the second piece and the third piece equal to each other at x = 3.

For x = 2, we have x² - 4x - 2 = a(2)² - b(2)³

For x = 3, we have ax² - bx³ = 4(3) - ab

Solving these equations will give us the values of a and b that make the function continuous.