Question

For which functions can the extreme value theorem be applied? check all that apply.

1) f(x) = 1/x over the interval [-2, 0]
2) f(x) = 1/(x - 3) over the interval [4, 10]
3) f(x) = x² over the interval (0, 2)
4) f(x) = (x - 2)³ over the interval (-10, -4]
5) f(x) = x⁴/(x² - 16) over the interval [-3, 3]

Answer 1

The functions where the Extreme Value Theorem can be applied are: f(x) = 1/(x - 3), f(x) = x², and f(x) = (x - 2)³.

Step-by-step explanation:

The Extreme Value Theorem states that for a continuous function f(x) over a closed interval [a, b], there will always be both a maximum and a minimum value of f(x) on that interval. Therefore, we need to check which of the given functions are continuous over their respective intervals.

1. f(x) = 1/x is not continuous at x = 0, so the Extreme Value Theorem cannot be applied to it over the interval [-2, 0].
2. f(x) = 1/(x - 3) is continuous over the interval [4, 10], so the Extreme Value Theorem can be applied to it.
3. f(x) = x² is continuous over the interval (0, 2), so the Extreme Value Theorem can be applied to it.
4. f(x) = (x - 2)³ is continuous over the interval (-10, -4], so the Extreme Value Theorem can be applied to it.
5. f(x) = x⁴/(x² - 16) is not continuous at x = ±4, so the Extreme Value Theorem cannot be applied to it over the interval [-3, 3].

Therefore, the functions where the Extreme Value Theorem can be applied are:

1. f(x) = 1/(x - 3) over the interval [4, 10]
2. f(x) = x² over the interval (0, 2)
3. f(x) = (x - 2)³ over the interval (-10, -4]