Find maximum of a polynomial? How would I find the value of p which the function f is the maximum:
$f(p)= 100 p(1-p)^{99}$  $p\geq0$
And in general, can setting the derivative equal to $0$ and solving for the variable lead to the values in function's domain that produces the maximum value ?

Here is another way;  clearly we seek $p \in (0, 1)$, so $p, 1-p$ are both positive.
$$\frac{99}{100}f(p) = (99p) \cdot \underbrace{(1-p)(1-p)\cdots(1-p)}_{99 \text{ times}} $$
Now the RHS is the product of $100$ terms, which sum to a constant, viz. $99$.  Therefore its maximum is when all terms are equal, viz. $\dfrac{99}{100}$ (this can be shown easily with AM-GM).  $\implies p = \dfrac1{100}$.
