minima maxima

I was given the problem
500t/(2t^2 + 9) and told
to find the maximum.

This can be done by the following:
500t/(2t^2 + 9) = k
By letting k represent the maximum
we get:
500t = 2*k*t^2 + 9*k
0 = 2*k*t^2 - 500t + 9*k

Then, given the discriminant
(-500)^2 - 4(2*k*9*k)
We get:
-72*k^2 + (-500)^2

And solving for k gives us
the maximum value.

My question is this,
why does it give the maximum
value? Why not some other value?
What is the relationship here?
That is, there is some underlying
relationship between the equation
(500t)/(2t^2 + 9) and it yielding the
maximum value. Which we don't know,
so we symbolize with k and thus solve
for k. But how do we know this equation
would yield the maximum value? What
is the underlying connection here?

I hope that is clear. Because I really
want to understand this.

--
conrad

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