What is the maximum value that the graph of ${sin}^{4} x + {cos}^{4} x$ $sin^4x+cos^4x$ ?

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What is the maximum value that the graph of ${sin}^{4} x + {cos}^{4} x$ $sin^4x+cos^4x$ ?

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Answer 1 · 2016-04-20T18:09:20

The maximum value is $1$ $1$

Explanation:

$f (x) = {sin}^{4} x + {cos}^{4} x = {({sin}^{2} x + {cos}^{2} x)}^{2} - 2 {sin}^{2} x {cos}^{2} x$ $f(x) = sin^4x+cos^4x = (sin^2x+cos^2x)^2-2sin^2xcos^2x$

$= 1 - 2 {sin}^{2} x {cos}^{2} x$ $= 1-2sin^2xcos^2x$

Notice that $2 {sin}^{2} x {cos}^{2} x$ $2sin^2xcos^2x$ cannot be negative, so

$2 {sin}^{2} x {cos}^{2} x \geq 0$ $2sin^2xcos^2x >= 0$ , and

$- 2 {sin}^{2} x {cos}^{2} x \leq 0$ $-2sin^2xcos^2x <= 0$ . So,

$1 - 2 {sin}^{2} x {cos}^{2} x \leq 1$ $1-2sin^2xcos^2x <= 1$

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

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Explanation:

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