Chapter 5.3, Problem 91E

Phytoplankton Growth Researchers have described the maximum growth rate of phytoplankton as a function of the cell diameter x by the equation

$g\left(x\right)=\frac{x}{a{x}^2+bx+c}$

wherea, b and c are positive constants and $x>0$.Source: The American Naturalist. For simplicity, let $a=b=c=1$.

a. The researchers state that when x is small, $g^{\prime }\left(x\right)>0$, but when x is large, $g^{\prime }\left(x\right)<0$. Find any intervals for which $g^{\prime }\left(x\right)>0$, any for which $g^{\prime }\left(x\right)<0$.

b.Calculate $g^{″}\left(x\right)$.

c. The researchers state that when x is small, $g^{″}\left(x\right)<0$, but when x is large, $g^{″}\left(x\right)>0$. Using a graphing calculator, find approximate intervals for which $g^{″}\left(x\right)<0$, and any for which $g^{″}\left(x\right)>0$.