Prove this statement by any method \"Any logistic curve that starts at 0 < P(0)

Question

Prove this statement by any method

"Any logistic curve that starts at 0 < P(0) < M/2 has its inflection point at the level M/2 That is, before the M/2 level is reached the velocity and the acceleration of growth have the same positive sign: an accelerated growth. After the M/2 level is reached, the velocity is still positive (still growth, not decay), but the acceleration P becomes negative: a decelerated growth."

I think the easiest way to prove it would be to differentiate the rate equation using the chain rule knowing that:

dP/dt = KP-kP^2 = kP(M-P)

Thanks!

Explanation / Answer

"Any logistic curve that starts at 0 < P(0) < M/2 has its inflection point at the level M/2 That is, before the M/2 level is reached the velocity and the acceleration of growth have the same positive sign: an accelerated growth. After the M/2 level is reached, the velocity is still positive (still growth, not decay), but the acceleration P becomes negative: a decelerated growth.

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