Finding when quotient rule can't be used

Question

I have the following problem:

My thought process has led to the following conclusions:

1. $f(x)$ must be a horizontal line, so $f'(x) = 0$
2. Due to $f'(x) = 0$, $h'(x) = 0$, so $h(x)$ is also a horizontal line.

From here, I was not able to make any other deductions, so I looked at the answer:

The answer is E) $2$ because no quotient rule so $g(x)$ must be non-zero constant.

I don't understand why quotient rule can't be used here. Can someone explain this vague answer?

Answer 1

I don't see how you deduced that $f(x)$ must be constant. But the answer is $2$ since the quotient rule says that $h'(x)=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g^2(x)}$. But it is given that this expression equals to $\frac{f'(x)}{g(x)}$. That implies that $g'(x)=0$ so $g(x)$ is constant. Since $g(5)=2$, we get $g(x)=2$ for all $x$.

Answer 2

You are given that $\frac{f'g-fg'}{g^2}=g'=\frac{f'}{g}$.

Therefore

$$-\frac{fg'}{g^2}=0$$

Since $f$ is never zero as implied by $f<0$, it must be that $g'=0$.

Therefore $g$ is constant. Since you are given the value $g(5)=2$, then $g(x)=2$ for all $x$.