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 $f(x) = 1$ .

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