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Consigo pensar em 3 formas para resolver.

1ª forma : Mínimo de uma parábola

[tex]\displaystyle \sf \text{sabemos que} : \\\\ sen^2(x)+cos^2(x) = 1 \\\\\ cos^2(x) = 1-sen^2(x) \\\\\\\ \text{queremos o m\'inimo de } :\\\\ M = sen^4(x)+cos^4(x) \\\\ \text{Fa\c ca} :\\\\ M = sen^4(x)+\left[cos^2(x)\right]^2 \\\\\ M = sen^4(x)+\left[1-sen^2(x)\right]^2 \\\\\ M = sen^4(x) +1-2sen^2(x)+sen^4(x) \\\\ M = 2sen^4(x)-2sen^2(x)+1\\\\ \text{note q M \'e uma par\'abola na vari\'avel }sen^2(x).\\\\ Assim,[/tex]

[tex]\displaystyle \sf \text{m\'inimo } =\frac{-\Delta}{4a} \\\\ \text{m\'inimo }=\frac{-((-2)^2-4\cdot 2\cdot 1)}{4\cdot 2}\\\\\ \text{m\'inimo }=\frac{-(4-8)}{4\cdot 2} \\\\\ \text{m\'inimo }=\frac{\not 4}{\not 4\cdot 2}=\frac{1}{2}\\\\\ \large\boxed{\begin{matrix}\text{Portanto o valor m\'inimo : } \\\\ \displaystyle \sf sen^4(x)+cos^4(x)=\frac{1}{2}\end{matrix}\ }\checkmark[/tex]

2ª forma : Desigualdade das médias

[tex]\displaystyle \sf \text{Pela desigualdade das m\'edias sabemos que } : \\\\ \sqrt{\frac{(a_1)^2+(a_2)^2+..+(a_n)^2}{n}}\geq \frac{a_1+a_2+..+a_n}{n}\\\\\ \text{Fa\c ca} : \\\\ a_1 = sen^2(x) \ ; \ a_2=cos^2(x)\\\\ Assim, \\\\\ \sqrt{\frac{(sen^2(x))^2+(cos^2(x))^2}{2}}\geq \frac{\overbrace{\sf sen^2(x)+cos^2(x)}^{\displaystyle 1}}{2} \\\\\\ \sqrt{\frac{sen^4(x)+cos^4(x)}{2}} \geq \frac{1}{2} \\\\\ \left(\sqrt{\frac{sen^4(x)+cos^4(x)}{2}} \right)^2\geq \left(\frac{1}{2} \right)^2[/tex]

[tex]\displaystyle \sf \frac{sen^4(x)+cos^4(x)}{2} \geq \frac{1}{4} \\\\\\ sen^4(x)+cos^4(x) \geq \frac{2}{4} \\\\\\\ sen^4(x)+cos^4(x)\geq \frac{1}{2}\\\\\\ \large\boxed{\begin{matrix}\text{valor m\'inimo} : \\\\ \displaystyle \sf \ sen^4(x)+cos^4(x) = \frac{1}{2} \ \end{matrix}}\checkmark[/tex]

3ª forma : Desigualdade de cauchy

[tex]\displaystyle \sf \text{usando a desigualdade de cauchy schwarz } : \\\\ (a_1\cdot b_1+a_2\cdot b_2) ^2\leq \left[(a_1)^2+(a_2)^2\right]\cdot \left[(b_1)^2+(b_2)^2\right] \\\\\\ \text{queremos} : \\\\ \left[sen^4(x)+cos^4(x)\right]_{\displaystyle \text{m\'inimo}}=\ ?\\\\ \text{Fa\c ca} : \\\\ \underbrace{\sf \left(1\cdot sen^2(x)+1\cdot cos^2(x)\right}_{\displaystyle 1})^2\leq (1^2+1^2) \cdot \left[(sen^2(x))^2+(cos^2(x))^2\right][/tex]

[tex]\displaystyle \sf 1\leq (2)\cdot \left[sen^4(x)+cos^4(x)\right] \\\\\\ sen^4(x)+cos^4(x) \geq \frac{1}{2} \\\\\\\ \large\boxed{\begin{matrix}\text{valor m\'inimo} : \\\\ \displaystyle \sf \ sen^4(x)+cos^4(x) = \frac{1}{2} \ \end{matrix}}\checkmark[/tex]