Respuesta :
Answer:
A. [tex]g'(3)=-\frac{1}{2}[/tex]
Step-by-step explanation:
We have been given that f be a differentiable function such that[tex]f(3)=15[/tex], [tex]f(6)=3[/tex], [tex]f'(3)= -8[/tex], and [tex]f'(6)= -2[/tex]. The function g is differentiable and [tex]g(x)= f^{-1} (x)[/tex] for all x.
We know that when one function is inverse of other function, so:
[tex]g(f(x))=x[/tex]
Upon taking derivative of both sides of our equation, we will get:
[tex]g'(f(x))*f'(x) =1[/tex]
[tex]g'(f(x))=\frac{1}{f'(x)}[/tex]
Plugging [tex]x=6[/tex] into our equation, we will get:
[tex]g'(f(6))=\frac{1}{f'(6)}[/tex]
Since [tex]g(x)= f^{-1} (x)[/tex], then [tex]g'(f(6))=g'(3)[/tex].
[tex]g'(3)=\frac{1}{f'(6)}[/tex]
Since we have been given that [tex]f'(6)= -2[/tex], so we will get:
[tex]g'(3)=\frac{1}{-2}[/tex]
[tex]g'(3)=-\frac{1}{2}[/tex]
Therefore, [tex]g'(3)=-\frac{1}{2}[/tex] and option A is the correct choice.
