Consider the function

$$f(x)=\sqrt{x+8}+\frac{1}{\sqrt{x+8}}$$

for $$x > -8$$. After differentiating, one finds that

$$f'(x)=\frac{1}{2*\sqrt{x+8}}-\frac{1}{2*(x+8)^{3/2}}.$$

Solve the equation $$f'(x)=0$$ to determine the critical number.