Question

If ="f(x)= \frac{1}{x-2} " align="absmiddle" class="latex-formula"> and (f*g)'(1)=6 and g'(1)=-1, then g(1)=
A. 5
B. -7
C. 7
D. 5
E. 8
(note: (f*g)'(1) means the derivitive of f(x)g(x) at x=1)
wow, the answer is 5, but why?
ples show all work and logic (don't just refernce to an online solver)

Answer 1

I think, the answer will be -7

We have:
f(x)=1/(x-2)
g(x)
Then:
(fg)(x)=[1/(x-2)](g(x))=g(x)/(x-2)
Now; we calculate: (fg)`(x)
Remember: (u/v)=(u`v-vu´)/v²
Therefore:

(fg)´(x)=[g´(x)*(x-2) - 1*g(x)]/ (x-2)²
We know that:
g´(1)=-1
(fg)´(1)=6
Therefore:
6=[-1*(1-2)-g(1)]/(1-2)²
6=[1-g(1)]/1
6=1-g(1)
-g(1)=6-1
g(1)=-5

Answer: B. -5