Question

Question 10(Multiple Choice Worth 10 points)
(03.03 MC)

Select which function f has an inverse g that satisfies g prime of 2 equals 1 over 6 period

f(x) = 2x3
f of x equals 1 over 8 times x cubed
f(x) = x3
1 over 3 times x cubed

Answer 1

$f(x) = 2\cdot x^{3}$ has an inverse $g$ that satisfies $g'(2) = \frac{1}{6}$.

Procedure - Inverse of a function

First, we determine the value of $x$ for $f(x)$ in each case, based on the fact that $f(x) = 2$:

Case A

$2\cdot x^{3} = 2$

$x = 1$

Case B

$2 = \frac{1}{8}\cdot x^{3}$

$16=x^{3}$

$x = \sqrt[3]{16}$

Case C

$2 = x^{3}$

$x = \sqrt[3]{2}$

Case D

$2 = \frac{1}{3}\cdot x^{3}$

$6 = x^{3}$

$x = \sqrt[3]{6}$

Now, we proceed to switch variables and find the derivative of the inverse function, whose formula is described below:

$g'(x) = \frac{1}{f'(y)}$ (1)

Where:

• $y$ - Dependent variable of the inverse function (Independent variable of the original function).
• $x$ - Independent variable of the inverse function (Dependent variable of the original function).

Case A ($x = 2$, $y = 1$)

$f(y) = 2\cdot y^{3}$

$f'(y) = 6\cdot y^{2}$

$f'(1) = 6\cdot 1^{2}$

$f'(1) = 6$

$g'(2) = \frac{1}{6}$ $\blacksquare$

Case B ($x = 2$, $y = \sqrt[3]{16}$)

$f(y) = \frac{1}{8}\cdot y^{3}$

$f'(y) = \frac{3}{8}\cdot y^{2}$

$f'(\sqrt[3]{16}) = \frac{3}{8}\cdot \sqrt[3]{4}$

$g'(2) = \frac{8}{3\cdot \sqrt[3]{4}}$

$g'(2) = \frac{8\cdot \sqrt[3]{16}}{12}$ $\blacksquare$

Case C ($x = 2$, $y = \sqrt[3]{2}$)

$f(y) = y^{3}$

$f'(y) = 3\cdot y^{2}$

$f'(\sqrt[3]{2}) = 3\cdot \sqrt [3]{4}$

$g'(2) = \frac{1}{3\cdot \sqrt [3]{4}}$

$g'(2) = \frac{\sqrt [3]{16}}{12}$ $\blacksquare$

Case D ($x = 2$, $y = \sqrt[3]{6}$)

$f(y) = \frac{1}{3}\cdot y^{3}$

$f'(y) = y^{2}$

$f(\sqrt[3]{6}) = \sqrt[3]{36}$

$g'(2) = \frac{1}{\sqrt [3]{36}}$

$g'(2) = \frac{36^{2/3}}{36}$

$g'(2) = \frac{\sqrt[3]{1296}}{36}$ $\blacksquare$

In a nutshell, $f(x) = 2\cdot x^{3}$ has an inverse $g$ that satisfies $g'(2) = \frac{1}{6}$. $\blacksquare$

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Remarks

The statement is poorly formated, correct form is presented below:

Select which function $f$ has an inverse $g$ that satisfies $g'(2) = \frac{1}{6}$:

A. $f(x) = 2\cdot x^{3}$

B. $f(x) = \frac{1}{8}\cdot x^{3}$

C. $f(x) = x^{3}$

D. $f(x) = \frac{1}{3}\cdot x^{3}$