This is poorly worded. I will assume "the width of the beam" is actually the

distance between the roots of the quadratic:

The roots of this are:

,

and the distance between the roots is:

Then:

Putting and , gives:

which does not agree with your book.

The possibilities are: that I have interpreted the problem incorrectly, I have

made a mistake in my algebra and derivatives, I have made a mistake in the

arithmetic, the book is wrong. No doubt my fellow helpers will comment on

the first three of these.

RonL