Question

Please help me with 2b ASAP. Really appreciate it!!

Answer 1

$f(x)=\dfrac{x^2}{x^2+k^2}$

By definition of the derivative,

$f'(x)=\displaystyle\lim_{h\to0}\frac{\frac{(x+h)^2}{(x+h)^2+k^2}-\frac{x^2}{x^2+k^2}}h$

$f'(x)=\displaystyle\lim_{h\to0}\frac{(x+h)^2(x^2+k^2)-x^2((x+h)^2+k^2)}{h(x^2+k^2)((x+h)^2+k^2)}$

$f'(x)=\dfrac{k^2}{x^2+k^2}\displaystyle\lim_{h\to0}\frac{(x+h)^2-x^2}{h((x+h)^2+k^2)}$

$f'(x)=\dfrac{k^2}{x^2+k^2}\displaystyle\lim_{h\to0}\frac{2xh+h^2}{h((x+h)^2+k^2)}$

$f'(x)=\dfrac{k^2}{x^2+k^2}\displaystyle\lim_{h\to0}\frac{2x+h}{(x+h)^2+k^2}$

$f'(x)=\dfrac{2xk^2}{(x^2+k^2)^2}$

$\dfrac{k^2}{(x^2+k^2)^2}$ is positive for all values of $x$ and $k$. As pointed out, $x\ge0$, so $f'(x)\ge0$ for all $x\ge0$. This means the proportion of occupied binding sites is an increasing function of the concentration of oxygen, meaning the presence of more oxygen is consistent with greater availability of binding sites. (The question says as much in the second sentence.)