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Sagot :
[tex]f(x) = 2\cdot x^{3}[/tex] has an inverse [tex]g[/tex] that satisfies [tex]g'(2) = \frac{1}{6}[/tex].
Procedure - Inverse of a function
First, we determine the value of [tex]x[/tex] for [tex]f(x)[/tex] in each case, based on the fact that [tex]f(x) = 2[/tex]:
Case A
[tex]2\cdot x^{3} = 2[/tex]
[tex]x = 1[/tex]
Case B
[tex]2 = \frac{1}{8}\cdot x^{3}[/tex]
[tex]16=x^{3}[/tex]
[tex]x = \sqrt[3]{16}[/tex]
Case C
[tex]2 = x^{3}[/tex]
[tex]x = \sqrt[3]{2}[/tex]
Case D
[tex]2 = \frac{1}{3}\cdot x^{3}[/tex]
[tex]6 = x^{3}[/tex]
[tex]x = \sqrt[3]{6}[/tex]
Now, we proceed to switch variables and find the derivative of the inverse function, whose formula is described below:
[tex]g'(x) = \frac{1}{f'(y)}[/tex] (1)
Where:
- [tex]y[/tex] - Dependent variable of the inverse function (Independent variable of the original function).
- [tex]x[/tex] - Independent variable of the inverse function (Dependent variable of the original function).
Case A ([tex]x = 2[/tex], [tex]y = 1[/tex])
[tex]f(y) = 2\cdot y^{3}[/tex]
[tex]f'(y) = 6\cdot y^{2}[/tex]
[tex]f'(1) = 6\cdot 1^{2}[/tex]
[tex]f'(1) = 6[/tex]
[tex]g'(2) = \frac{1}{6}[/tex] [tex]\blacksquare[/tex]
Case B ([tex]x = 2[/tex], [tex]y = \sqrt[3]{16}[/tex])
[tex]f(y) = \frac{1}{8}\cdot y^{3}[/tex]
[tex]f'(y) = \frac{3}{8}\cdot y^{2}[/tex]
[tex]f'(\sqrt[3]{16}) = \frac{3}{8}\cdot \sqrt[3]{4}[/tex]
[tex]g'(2) = \frac{8}{3\cdot \sqrt[3]{4}}[/tex]
[tex]g'(2) = \frac{8\cdot \sqrt[3]{16}}{12}[/tex] [tex]\blacksquare[/tex]
Case C ([tex]x = 2[/tex], [tex]y = \sqrt[3]{2}[/tex])
[tex]f(y) = y^{3}[/tex]
[tex]f'(y) = 3\cdot y^{2}[/tex]
[tex]f'(\sqrt[3]{2}) = 3\cdot \sqrt [3]{4}[/tex]
[tex]g'(2) = \frac{1}{3\cdot \sqrt [3]{4}}[/tex]
[tex]g'(2) = \frac{\sqrt [3]{16}}{12}[/tex] [tex]\blacksquare[/tex]
Case D ([tex]x = 2[/tex], [tex]y = \sqrt[3]{6}[/tex])
[tex]f(y) = \frac{1}{3}\cdot y^{3}[/tex]
[tex]f'(y) = y^{2}[/tex]
[tex]f(\sqrt[3]{6}) = \sqrt[3]{36}[/tex]
[tex]g'(2) = \frac{1}{\sqrt [3]{36}}[/tex]
[tex]g'(2) = \frac{36^{2/3}}{36}[/tex]
[tex]g'(2) = \frac{\sqrt[3]{1296}}{36}[/tex] [tex]\blacksquare[/tex]
In a nutshell, [tex]f(x) = 2\cdot x^{3}[/tex] has an inverse [tex]g[/tex] that satisfies [tex]g'(2) = \frac{1}{6}[/tex]. [tex]\blacksquare[/tex]
To learn more on inverse functions, we kindly invite to check this verified question: https://brainly.com/question/5245372
Remarks
The statement is poorly formated, correct form is presented below:
Select which function [tex]f[/tex] has an inverse [tex]g[/tex] that satisfies [tex]g'(2) = \frac{1}{6}[/tex]:
A. [tex]f(x) = 2\cdot x^{3}[/tex]
B. [tex]f(x) = \frac{1}{8}\cdot x^{3}[/tex]
C. [tex]f(x) = x^{3}[/tex]
D. [tex]f(x) = \frac{1}{3}\cdot x^{3}[/tex]
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