By Fundamental Theorem of Calculus,

Since  is a constant, its derivative is . Therefore, 

To begin with write the information of the problem in mathematical terms. We're told the sum of two numbers is fifty:

Their product is then:

We're uncertain currently about what these two numbers are, but we can relate them:

So that we can simplify the product function:

Maxima and minima occur where the derivative of a function is zero. Take the derivative of this function:

And set it to zero:

You can validate that this is a minimum due to the derivative function having positive values before this point, and negative values after this point.

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.

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of

 on the interval 

Then take the difference of the two and divide by the interval.

Now find the derivative of the function; this will be solved for the value(s) found above.

This value valls within the interval , validating the mean value theorem.

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.

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of

 on the interval 

Then take the difference of the two and divide by the interval.

Now find the derivative of the function; this will be solved for the value(s) found above.

To validate the mean value theorem, a solution must fall within the interval used, and  does in fact fall within 

Use the power rule to find the derivative.

Thus, the derivative is 

Use the power rule to find the derivative. 

First, find the derivative using the power rule:

Now, substitute 3 for x.

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.

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of

 on the interval 

Then take the difference of the two and divide by the interval.

Now find the derivative of the function; this will be solved for the value(s) found above.

Derivative of an exponential: 

Trigonometric derivative: 

Using a solver, there are multiple solutions which satisfy this equation:

However, to satisfy the mean value theorem, a solution must fall within the specified interval.  falls within .

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 interval and function definitions, along with the derivative value of  given, we'll in turn wish to solve for  to validate the mean value theorem:

(Note that the solution  would not work because it would eliminate the interval as , and it would lead to a zero denominator in the working equation.)