Final answer:
To compute f-1'(x) at x = -3 for an increasing function f with given values f(1) = -3 and f '(1) = 3, we use the theorem for the derivative of an inverse function. The result is f-1'(-3) = 1 / f '(1) = 1 / 3, indicating option c. 1 is the correct answer.
Step-by-step explanation:
The question concerns the computation of the derivative of the inverse function, f-1'(x). The information given states that for the function f, f(1) = −3 and f '(1) = 3. According to the theorem for the derivative of an inverse function, if f is differentiable at a point b, and f ' at b is not equal to zero, then the inverse function f−1 is differentiable at f(b) and:
f−1'(f(b)) = 1 / f '(b)
Applying this to the given information:
f−1'(−3) = 1 / f '(1) = 1 / 3
Given that function f is an increasing function and satisfies f(1) = -3 and f'(1) = 3, we can find f-1'(-3) by finding the derivative of the inverse function at x = -3. The derivative of the inverse function is equal to the reciprocal of the derivative of the original function at the corresponding point.
So, we need to find f'(1). Since f(x) is an increasing function, f'(x) is positive for all x. Therefore, f'(1) = 3.
Therefore, the answer to the question is c. 1.