A decreasing function g satisfies g(4) = 6 and g'(4) = -2. Which of the following statements about the inverse of g must be true? a. (g^-1)'(6) = -2
b. (g^-1)'(-2) = 4
c. (g^-1)'(6) = -2
d. (g^-1)'(6) = -1/2

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.

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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.

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