Explanation Given: The function is shown below. y = x p ( 1 − x ) q ...... (1) Calculation: The given function is an exponential function. To obtain the critical point of the function, differentiate the equation (1) as below. d y d x = d d x x p ( 1 − x ) q y ′ = ( 1 − x ) q d d x x p + x p d d x ( 1 − x ) q y ′ = ( 1 − x ) q p x p − 1 + x p ⋅ q ( 1 − x ) q − 1 ⋅ ( − 1 ) y ′ = x p ( 1 − x ) q ( p x − q 1 − x ) To obtain the critical point equate the function y ′ to 0. x p ( 1 − x ) q ( p x − q 1 − x ) = 0 Use the zero product property to obtain the value of x . The first value of x is calculated as shown below: x p = 0 x = 0 The second value of x is calculated as shown below: ( 1 − x ) q = 0 1 − x = 0 x = 1 The third value of x is calculated as shown below: ( p x − q 1 − x ) = 0 p x = q 1 − x p ( 1 − x ) = q x p − p x = q x Simplify the above equation as follows: p = x ( q + p ) x = p p + q Therefore, the critical points are 0 , 1 and p p + q

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 4.2, Problem 100E

To determine

To find:

The maximum and minimum value of the function y=xp(1−x)q on the interval [0,1] and p,q>0.

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Chapter 4.2, Problem 99E

Chapter 4.2, Problem 101E

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Chapter 4 Solutions

Calculus: Early Transcendentals

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The maximum and minimum value of the function y = x p ( 1 − x ) q on the interval [ 0 , 1 ] and p , q > 0 .

Solution Summary: The author explains that the given function is an exponential function. To obtain the critical point of the function, differentiate the equation as below.

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The maximum and minimum value of the function y=xp(1−x)q on the interval [0,1] and p,q>0.

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The maximum and minimum value of the function y = x p ( 1 − x ) q on the interval [ 0 , 1 ] and p , q &gt; 0 .

Solution Summary: The author explains that the given function is an exponential function. To obtain the critical point of the function, differentiate the equation as below.

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To find: The maximum and minimum value of the function y=xp(1−x)q on the interval [0,1] and p,q>0.

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Chapter 4 Solutions

The maximum and minimum value of the function y = x p ( 1 − x ) q on the interval [ 0 , 1 ] and p , q > 0 .

To find:

The maximum and minimum value of the function y=xp(1−x)q on the interval [0,1] and p,q>0.