Answered

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An increasing function f satisfies f(10) = 5 and f'(10) = 8. Which of the following statements about the
inverse of f must be true?

(A). (f^-1)’(5) = 10
(B). (f^-1)’(8) = 10
(C). (f^-1)’(5) = 8
(D). (f^-1)’(5) = 1/8

Sagot :

Answer:  D

Explanation:

Because [tex]f^{-1}'(y)=\frac{1}{f'(x)}[/tex] and y=5 when x=10, [tex]f^{-1}'(5)=\frac{1}{f'(10)}=\frac{1}{8}[/tex]

MrRoyal

The given equations are illustrations of inverse functions. The statement about the inverse of f that must be true is: [tex]f^{-1}(5) = \frac{1}{8}[/tex]

Given that:

[tex]f(10) = 5[/tex]

[tex]f'(10) = 8[/tex]

[tex]f(10) = 5[/tex] means that:

[tex]x = 10[/tex] and [tex]y = 5[/tex]

So, the relationship between the inverse of both functions is:

[tex]f^{-1}(y) = \frac{1}{f^{-1}(x)}[/tex]

Substitute values for x and y

[tex]f^{-1}(5) = \frac{1}{f^{-1}(10)}[/tex]

Substitute [tex]f'(10) = 8[/tex]

[tex]f^{-1}(5) = \frac{1}{8}[/tex]

Hence, option D is true, i.e.

[tex]D.\ f^{-1}(5) = \frac{1}{8}[/tex]

Read more about inverse functions at:

https://brainly.com/question/10300045

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