IVT, EVT, MVT

Front

What is the Intermediate Value Theorem (IVT)?

Back

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], then for any value L between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = L.

Front

What is the Extreme Value Theorem (EVT)?

Back

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it attains both a maximum and a minimum value at some points in that interval.

Front

What is the Mean Value Theorem (MVT)?

Back

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

How do you determine if a function has a relative maximum or minimum at a point?

Back

To determine if a function has a relative maximum or minimum at a point, use the first derivative test: if f'(x) changes from positive to negative at x, then f has a relative maximum; if it changes from negative to positive, then f has a relative minimum.

Front

What is a continuous function?

Back

A continuous function is a function that does not have any breaks, jumps, or holes in its graph. Formally, a function f is continuous at a point x = c if lim (x -> c) f(x) = f(c).

Front

What is a differentiable function?

Back

A differentiable function is a function that has a derivative at each point in its domain. This means that the function is smooth and has no sharp corners or cusps.

Front

Provide an example of a function that satisfies the conditions of the Mean Value Theorem on the interval [1, 3].

Back

Consider the function f(x) = x^2. It is continuous on [1, 3] and differentiable on (1, 3). By MVT, there exists c in (1, 3) such that f'(c) = (f(3) - f(1)) / (3 - 1).

Front

What is the significance of the endpoints in the Extreme Value Theorem?

Back

In the Extreme Value Theorem, the endpoints of the interval are significant because the maximum and minimum values can occur at these endpoints or at critical points within the interval.

Front

What is a critical point?

Back

A critical point of a function f is a point x in its domain where f'(x) = 0 or f'(x) does not exist. Critical points are potential locations for relative maxima and minima.

Front

How can you apply the Intermediate Value Theorem to show that a root exists?

Back

To apply the Intermediate Value Theorem to show that a root exists, find two points a and b such that f(a) and f(b) have opposite signs. Then, by IVT, there exists at least one c in (a, b) where f(c) = 0.