To find which of the given decreasing functions satisfies the condition (f^-1)'(10) = -2, we need to calculate the derivative of the inverse function of each function and evaluate it at x = 10.
Let's go through each option:
A) f(x) = -5x + 15
To find the inverse function, we need to swap x and y and solve for y:
x = -5y + 15
5y = 15 - x
y = (15 - x)/5
Taking the derivative of the inverse function:
(f^-1)'(x) = -1/5
(f^-1)'(10) = -1/5 ≠ -2
B) f(x) = -223 - 2x + 14
To find the inverse function, we swap x and y:
x = -223 - 2y + 14
2y = -237 - x
y = (-237 - x)/2
Taking the derivative of the inverse function:
(f^-1)'(x) = -1/2
(f^-1)'(10) = -1/2 ≠ -2
C) f(x) = -25 - 4x + 15
To find the inverse function, we swap x and y:
x = -25 - 4y + 15
4y = -10 - x
y = (-10 - x)/4
Taking the derivative of the inverse function:
(f^-1)'(x) = -1/4
(f^-1)'(10) = -1/4 ≠ -2
D) f(x) = -21 - 2 + 9
This function seems to be incorrect since there is a missing variable. It should be written as f(x) = -21x - 2x + 9.
To find the inverse function, we swap x and y:
x = -21y - 2y + 9
-23y = 9 - x
y = (9 - x)/-23
Taking the derivative of the inverse function:
(f^-1)'(x) = -1/23
(f^-1)'(10) = -1/23 ≠ -2
None of the given functions satisfy the condition (f^-1)'(10) = -2.