The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  gives the solution:

Which is, as expected, greater than  to allow the creation of an interval.

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, ,  , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  gives the solution:

Which is, as expected, greater than  to allow the creation of an interval.

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  gives the solution:

Which is, as expected, greater than  to allow the creation of an interval.

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  via numerical solver gives the solution:

Choosing the positive solution for the sake of the interval leaves

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  using a calculator gives the solution:

Which is, as expected, greater than  to allow the creation of an interval.

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  using a calculator gives the solution:

Which is, as expected, greater than , albeit just barely, to allow the creation of an interval.

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  using a calculator gives the solution:

There are two solutions to the equation; elect the value larger than  to allow creation of an interval:

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  using a calculator gives the solution:

To enable the creation of an interval, elect the value greater than a:

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  using a calculator gives the solution:

Elect the value that is greater than a to complete the interval:

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Using our function, interval, and derivative definitions, , , , we'll in turn wish to solve for  to validate the mean value theorem:

Solving for  using a calculator gives the solution:

Which is, as expected, greater than  to allow the creation of an interval.