If f is a periodic function, then the locations of all absolute extrema on the interval (–∞,+∞)

Question

If f is a periodic function, then the locations of all absolute extrema on the interval

(–∞,+∞)

can be obtained by finding the locations of the absolute extrema for one period and using the periodicity to locate the rest. Use this idea in these exercises to find the absolute maximum and minimum values of f(x)=2 cos x+cos 2x and state the x-values at which they occur.

f has an absolute maximum of ?

f has an absolute minimum of?

Answer 1

f'(x)=-2sinx-2sin2x is zero whensin2x= -sinx=sin(-x)

again need to solve sin2x=sin(-x)

A.same as 2x= -x+2πk where k=0,+/-1,+/-2,...

B.and is also the same as 2x=π-(-x)+2πk where k=0,+/-1,+/-2,..

in case A. f'(x)=0 when x=2πk/3where k=0,+/-1,+/-2,... k=0,+/-1,+/-2,

these are all potential local min/max, note f''(x)=-2cosx-4cos2x

when k=0, f''<0 so that is a maximum, at k=1 and 2 f'';>0 so those two are minimums.

then is repeats with k=3

In case B, x=3πk where k=0,+/-1,+/-2,... which are repeats of values found in case A.

so min/max at x=2πk/3 where k=0,+/-1,+/-2,..., maximums at k=0,+/-3,+/-6 minimums at the other points

value of f(x) maximum is 3, minimum of f(x) is -1.5