Final answer:
To find the value of (F^-1)(2), use the Local Inverse Theorem; since f maps 6 to 2 with a derivative of 1, F^-1 maps 2 back to 6. With f'(6) = 1, the derivative of F^-1 at 2 is the reciprocal of 1, which is 1.
Step-by-step explanation:
The student's question is about finding the value of (F-1) (a), which implies we are looking for the value of the inverse function of 'f' evaluated at 'a'. Given f(6) = 2, f'(6) = 1, and a = 2, we are to infer the value of (F-1) (2). This is typically approached using the Local Inverse Theorem which states that if 'f' is a differentiable function with a non-zero derivative at a point 'b', then its inverse function F-1 is also differentiable at f(b). And the derivative of F-1 at f(b) is the reciprocal of the derivative of 'f' at 'b'.
Thus, since f'(6) = 1, we can infer that (F-1)'(2) = 1/1 = 1. If we want the value of (F-1)(2), we refer to the given value that f(6) = 2. Since 'f' maps 6 to 2, the inverse function F-1 must map 2 back to 6, so (F-1)(2) = 6. This answers part 290 of the student's question. Part 294 indicates f(1) = 0, f'(1) = -2, and a = 0 which would be solved using a similar methodology.