Flux integral using Divergence theorem

Question

I have done part a, but I am not sure about part b. I was thinking maybe linking the surface area of a sphere with radius $1$ $=4\pi$ with part a? But if this is correct I'm not sure why.

Answer 1

$$\frac{4\pi}{3} |H| \overset{a}{\le} \iint_{|x| = 1} \left|\frac{(\nabla f)\cdot x}{\sqrt{1 + |\nabla f|^2}}\right| dA \overset{b}{\le} \iint_{|x| = 1} \frac{|(\nabla f)||x|}{\sqrt{1 + |\nabla f|^2}} dA \overset{c}{\le} \iint_{|x| = 1} 1 dA = 4\pi $$

Where

Inequality $a$ follows from $|\int f(x) dx| \le \int |f(x)| dx$,

inequality $b$ from $|u \cdot v| = | |u| |v| \cos \theta| \le |u||v|$, and

inequality $c$ from $|x| = 1$, and $\frac{u}{\sqrt{1 + u^2}}\le 1$