Find the absolute extrema of the function on the closed interval.
[tex]y=9 \cos (x), \quad[0,2 \pi][/tex]
minimum
(x, y)=(\square)
maximum (smaller [tex]x[/tex]-value) (x, y)=(\square)
maximum (larger, [tex]x[/tex]-value)
(x, y)=(\square)
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Answer (1)
Find the derivative of the function y = 9 cos ( x ) , which is d x d y = − 9 sin ( x ) .
Set the derivative to zero to find critical points: − 9 sin ( x ) = 0 , which gives x = 0 , π , 2 π in the interval [ 0 , 2 π ] .
Evaluate the function at the critical points and endpoints: y ( 0 ) = 9 , y ( π ) = − 9 , y ( 2 π ) = 9 .
Identify the absolute minimum and maximum: Minimum is ( π , − 9 ) , and maxima are ( 0 , 9 ) and ( 2 π , 9 ) .
( π , − 9 ) ( 0 , 9 ) ( 2 π , 9 )
Explanation
Problem Analysis We are given the function y = 9 cos ( x ) and the interval [ 0 , 2 π ] . Our goal is to find the absolute maximum and minimum values of the function on this interval.
Finding Critical Points First, we need to find the critical points of the function. To do this, we find the derivative of y with respect to x :
d x d y = − 9 sin ( x ) We set the derivative equal to zero to find the critical points: − 9 sin ( x ) = 0 sin ( x ) = 0 The solutions to this equation in the interval [ 0 , 2 π ] are x = 0 , x = π , and x = 2 π .
Evaluating the Function Now, we evaluate the function at the critical points and the endpoints of the interval:
At x = 0 :
y = 9 cos ( 0 ) = 9 At x = π :
y = 9 cos ( π ) = − 9 At x = 2 π :
y = 9 cos ( 2 π ) = 9
Identifying Extrema From the evaluated values, we can identify the absolute maximum and minimum values:
The absolute maximum value is 9 , which occurs at x = 0 and x = 2 π .
The absolute minimum value is − 9 , which occurs at x = π .
Final Answer Therefore, the absolute extrema are:
Minimum: ( π , − 9 ) Maximum (smaller x -value): ( 0 , 9 ) Maximum (larger x -value): ( 2 π , 9 )
Examples
Imagine you're designing a sound wave using a cosine function. Knowing the absolute maximum and minimum values helps you determine the wave's amplitude, which directly affects the loudness of the sound. Understanding extrema is crucial for controlling and predicting the behavior of waves in various applications, from audio engineering to physics.
