Let be a function on the interval that is nonnegative and differentiable, and such that . If for each in the open interval the line given by the equation divides the ordinate set of into regions and where denotes the leftmost region. If the areas of the two regions obey the equation

where is a constant that does not depend on , find and .

**Incomplete.**

First, let’s define the regions A and B as functions of the variable a

Now, we take our given equation:

If we take the derivative with respect to a of both sides of the equation, we get

This equation is satisfied by the function

Referring back to our initial condition that f(1) = 0, we can solve for the constant C

As requested.

And to find the scalar constant b, all we need to do is use our newly found function and evaluate the above integrals.

Recall,

And we now now that the unknown function f(x):

The above equation becomes