Little bit of help please?

Subbing x for t......look at the graph here, Julius

https://www.desmos.com/calculator/6wqntzjcax

The graph has a positive slope from [0, 3]....so the max is reached at 3 months ≈ 512.733 (thousands)

I see that. But how do you do it algebraically?

Wouldn't you just sub 3 for for t in your given function? Try it:

P(3) =4*3 + (100*3 / 3^2+400) + 500

P(3) = 12 + (300 / 409) + 500

P(3) = 12 + 0.733496... + 500

P(3) = 512.733486......

True: But w/o seeing the graph how would you know that at 3 months it's at its maximum population?