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For which of the following decreasing functions f does (f^-1)'(10) = -2 А f(x) = -5x + 15 B f(x) = -223 - 2x + 14 C f(x) = -25 - 4x + 15 D f(x) = -21 - 2 +9

User Aeh
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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.

User Eric Cornelson
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