SOLUTION: Find the maximum and minimum values of each given function and state the corresponding values of x . ( 0 <= x < 2pi ) (1) {{{y=cos(x)+2cos(x/2)}}}

Algebra.Com
Question 1047187: Find the maximum and minimum values of each given function and state the corresponding values of x . ( 0 <= x < 2pi )
(1)
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
Find the maximum and minimum values of each given function and state the corresponding values of x. ( 0 <= x < 2pi )
(1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Take the derivative on "x": y'(x) = -sin(x) + 2*(-sin(x/2)*(1/2) = - sin(x) - sin(x/2).

Equate it to zero. You will get an equation

-sin(x) - sin(x/2) = 0,   or

2*sin(x/2)*cos(x/2) + sin(x/2) = 0,   or   ( after factoring )

sin(x/2)*(2*cos(x/2) + 1) = 0.

It deploys in two independent equations


1.  sin(x/2) = 0  --->  x/2 = 0, ,   --->  x = 0  and  x =   ( the only roots in the segment [,] ).


2.  2*cos(x/2) + 1 = 0  --->  cos(x/2) =   ---> x/2 = ,   --->  x =   ( the only root in the segment [,] ).
 

Answer:

  a)  Maximum at x = 0:                        3.


  b)  Minimum at x = :     = -1.5.


  c)  Local maximum at x = : 1 + 2*(-1) = -1.



Plot y = cos(x)+2cos(x/2)


RELATED QUESTIONS

Find the maximum and minimum values of each given function and state the corresponding... (answered by ikleyn)
Graph two full periods of the function. f(x) = 2 cos(𝜋x) WebAssign Plot WebAssign... (answered by MathLover1)
Graph two full periods of the function. f(x) = 2 cos(𝜋x) State the amplitude,... (answered by MathLover1)
Graph one full period of the function, starting at x = 0. f(x) = 7 sin(x-(5pi)/6) State (answered by MathLover1)
Hi! Find the local maximum and minimum values of the function and the value of x at... (answered by solver91311)
Find the minimum and maximum values of the objective function subject to the given... (answered by Fombitz)
How do you find the minimum and maximum values of each function over the feasible region (answered by KMST)
What is the maximum and minimum value value of y=2sin(x)+2 and what are the corresponding (answered by stanbon)
What are the maximum and minimum values of objective function p=x+4y with given... (answered by Fombitz)