How do you find the critical numbers for #f(x) = x + 2sinx# to determine the maximum and minimum?

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Answer 1 · 2018-07-22T16:03:12

The critical points of a function #f(x)# are the #x# that make #f'(x)=0#

We calculate the derivative #f'(x)=1+2cos(x)#, and now we need to find where #f'(x)=1+2cos(x)=0#. But that means:

#-1=2cos(x)#, and then #cos(x) = -1/2#. Between #0# and #2pi# these points are:

#x=4/6pi=2/3pi# and #8/6pi=4/3pi#,

and all the congruents are:

#x=2/3pi+K*2pi# and #4/3pi+K*2pi# where #K# is any integer number (positive or negative)

Now the second derivative of #f(x)# is:

#f''(x) = -2sin(x)# which is negative in the first set of points and positive in the second set. So the points:

#x=2/3pi+K*2pi# are all local maxima, and the points
#4/3pi+K*2pi# are all local minima