What is the maximum value that the graph of #sin^4x+cos^4x#?

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Answer 1 · 2016-04-20T18:09:20

The maximum value is #1#

#f(x) = sin^4x+cos^4x = (sin^2x+cos^2x)^2-2sin^2xcos^2x#

# = 1-2sin^2xcos^2x#

Notice that #2sin^2xcos^2x# cannot be negative, so

#2sin^2xcos^2x >= 0#, and

#-2sin^2xcos^2x <= 0#. So,

#1-2sin^2xcos^2x <= 1#

Also note that for some #x#, we have #sinx = 0#, so for some #x#, we get #f(x) = 1#.