SOLUTION: use calculus to find the max and min of x - 2*cos(x) for the interval -2<= x <= 0

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Question 289780: use calculus to find the max and min of
x - 2*cos(x)
for the interval -2<= x <= 0
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
To find extrema, take the derivative and set it equal to zero.
f%28x%29=x-2%2Acos%28x%29
df%2Fdx=1-2%2A%28-sin%28x%29%29=1%2B2%2Asin%28x%29=0
1%2B2%2Asin%28x%29=0
2%2Asin%28x%29=-1
sin%28x%29=-1%2F2
+graph%28+300%2C+200%2C+-7%2C+1%2C+-10%2C+2%2C+x-2%2Acos%28x%29%29+
.
.
.
sin%28x%29=-1%2F2
x=-pi%2F6 and x=-5pi%2F6
From the graph, the maximum occurs at x=-5pi%2F6,
f%28x%29=-5pi%2F6-2%2Acos%28-5pi%2F6%29
f%28x%29=-5pi%2F6-2%2A%28-sqrt%283%29%2F2%29
f%28x%29=-5pi%2F6%2Bsqrt%283%29%29 or approximately
fmin=-0.886
.
.
.
As you can see from the graph, solving for x only gives you the relative minimum and not the absolute minimum.
The absolute minimum occur at the endpoints.
fmin=f%28-2%2Api%29=-2%2Api-2%2Acos%282%2Api%29=-2%2Api-2=-8.28