MVT, IVT, EVT

Front

What is the Mean Value Theorem (MVT)?

Back

The MVT 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 point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What is the Intermediate Value Theorem (IVT)?

Back

The IVT states that if a function is continuous on a closed interval [a, b], and N is any number between f(a) and f(b), then there exists at least one point c in (a, b) such that f(c) = N.

Front

What is the Extreme Value Theorem (EVT)?

Back

The EVT states that if a function is continuous on a closed interval [a, b], then it must attain a maximum and a minimum value at least once in that interval.

Front

Can the MVT be applied to the function f(x) = (-x^2 + 9)/(4x) on [1,3]?

Back

Yes, the MVT can be applied because the function is continuous and differentiable on the interval.

Front

Find all values of c that satisfy the MVT for f(x) = -x^2 + 8x - 17 on [2,6].

Back

c = 4.

Front

Find the value of c guaranteed by the IVT for f(x) = x^2 + x - 1 on [0,5] where f(c) = 11.

Back

c = 3.

Front

If f(x) is continuous on the interval [-2, 5] and f(-2) = 5 and f(5) = -2, what theorem applies?

Back

IVT.

Front

Does the EVT apply to f(x) = x^2/(3x - 6) on [3,6]?

Back

Yes, the EVT applies because the function is continuous on the interval.

Front

What conditions must be met for the MVT to apply?

Back

The function must be continuous on a closed interval [a, b] and differentiable on the open interval (a, b).

Front

What is a practical application of the IVT?

Back

The IVT can be used to show that a continuous function has a root in an interval if it takes on values of opposite signs at the endpoints.