Understanding the Mean Value Theorem

The function must be continuous on the open interval and differentiable on the closed interval.

The function must be continuous on the closed interval and differentiable on the open interval.

The function must be differentiable on both the open and closed intervals.

The function must be continuous on both the open and closed intervals.

The minimum value of the function.

The average rate of change over the interval.

The instantaneous rate of change at a point.

The maximum value of the function.

It is the point where the function reaches its maximum value.

It is the point where the tangent line is parallel to the secant line.

It is the point where the function is not differentiable.

It is the point where the function is not continuous.

It represents the tangent line to the function.

It represents the secant line to the function.

It represents the maximum value of the function.

It represents the minimum value of the function.

H(x) is the sum of f(x) and g(x).

H(x) is the product of f(x) and g(x).

H(x) is the difference between f(x) and g(x).

H(x) is the quotient of f(x) and g(x).

The Fundamental Theorem of Calculus

The Intermediate Value Theorem

The Extreme Value Theorem

Rolle's Theorem

The derivative of H(x) is equal to the slope of the secant line.

The derivative of H(x) is equal to zero.

The derivative of H(x) is equal to the derivative of f(x).

The derivative of H(x) is equal to the slope of the tangent line.

It guarantees multiple points C.

It guarantees no points C.

It guarantees at least one point C.

It guarantees exactly one point C.

Showing that f'(C) equals the slope of the secant line.

Showing that f'(C) equals the maximum value of the function.

Showing that f'(C) equals the minimum value of the function.

Showing that f'(C) equals zero.

The function is decreasing.

The function is constant.

The function is increasing.

The conditions of continuity and differentiability are met.