Answered: graph each function. Then use the…

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter9: Quadratic Functions And Equations

Section9.2: Transformations Of Quadratic Functions

Problem 43HP

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Question

graph each function. Then use the function’s first
derivative to explain what you see. y = x^1/3 + (x - 1)^1/3

Expert Solution

Step 1

Given that:

The equation is $y = x^{\frac{1}{3}} + {(x - 1)}^{\frac{1}{3}}$ .

Step 2

By using the formula,

$\frac{d}{d x} x^{n} = n x^{n - 1}$

If f^'( x ) > 0 ,then f ( x) is the increasing function.

Step 3

To find the first derivative of the function.

Differentiate the above equation with respect to x,

by using the above formula,

$y^{'} = \frac{d}{d x} (x^{\frac{1}{3}} + {(x - 1)}^{\frac{1}{3}}) = \frac{d}{d x} x^{\frac{1}{3}} + \frac{d}{d x} {(x - 1)}^{\frac{1}{3}} = \frac{1}{3} x^{- \frac{2}{3}} + \frac{1}{3} {(x - 1)}^{- \frac{2}{3}} = \frac{1}{3} (x^{- \frac{2}{3}} + {(x - 1)}^{- \frac{2}{3}}) = \frac{1}{3} (\frac{1}{x^{\frac{2}{3}}} + \frac{1}{{(x - 1)}^{\frac{2}{3}}}) \Rightarrow y^{'} = \frac{1}{3} (\frac{1}{x^{\frac{2}{3}}} + \frac{1}{{(x - 1)}^{\frac{2}{3}}})$

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