the diversity of a population that contains two species can be measured by the gini - simpson diversity index. if the fraction of organisms from species 1 is p, then the diversity is given by h(p) below. use the product rule to calculate h(p). for what value of p does h(p)=0?
h(p)=2p(1 - p)
h(p)=

Question

Accepted Answer

# Explanation:
## Step1: Apply the product rule
The product rule states that if $y = uv$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = 2p$ and $v=(1 - p)$. Then $u^\prime=2$ and $v^\prime=- 1$.
$$
\begin{align*}
H^\prime(p)&=(2p)^\prime(1 - p)+2p(1 - p)^\prime\
\end{align*}
$$
## Step2: Calculate the derivative
Substitute $u^\prime = 2$ and $v^\prime=-1$ into the product - rule formula.
$$
\begin{align*}
H^\prime(p)&=2(1 - p)+2p(-1)\
&=2-2p - 2p\
&=2-4p
\end{align*}
$$
## Step3: Solve $H^\prime(p)=0$
Set $H^\prime(p)=0$, so $2-4p = 0$.
$$
\begin{align*}
2-4p&=0\
-4p&=-2\
p&=\frac{1}{2}
\end{align*}
$$

# Answer:
$H^\prime(p)=2 - 4p$ and $p = \frac{1}{2}$ when $H^\prime(p)=0$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the product rule

Step2: Calculate the derivative

Step3: Solve \(H^\prime(p)=0\)

Answer: