The inverse function theorem (terms and conditions apply) says that, if f(a) = b, then
(f ⁻¹)'(b) = 1 / f '(a) = 1 / f '(f ⁻¹(b))
We're given that g (4) = 6, so from the theorem it follows that
(g ⁻¹)'(6) = 1 / g '(4) = -1/2
so only D must be true.
B might also be true, but we don't have enough information to determine that.
The statement that is true about the inverse of g from the given options is;
Option D; (g ⁻¹)'(6) = -1/2
The theorem for inverse functions is one that is stated by the relationship that if f(x) = y, then the derivative of the inverse function is given as;
(f ⁻¹)'(y) = 1/f'(x) = 1/f'(f ⁻¹(y))
From the question, we see that; g(4) = 6 and g'(4) = -2. Thus, from the theorem, we can see that;
(g ⁻¹)'(6) = 1/g'(4)
Plugging in -2 for g'(4) gives;
(g ⁻¹)'(6) = -1/2
Looking at the options, the only one that corresponds with our answer is Option D.
Read more at; https://brainly.in/question/3055490
