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

Apr 20, 2016

The maximum value is $1$

Explanation:

$f \left(x\right) = {\sin}^{4} x + {\cos}^{4} x = {\left({\sin}^{2} x + {\cos}^{2} x\right)}^{2} - 2 {\sin}^{2} x {\cos}^{2} x$

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

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

$2 {\sin}^{2} x {\cos}^{2} x \ge 0$, and

$- 2 {\sin}^{2} x {\cos}^{2} x \le 0$. So,

$1 - 2 {\sin}^{2} x {\cos}^{2} x \le 1$

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