jmc

By Cauchy-Schwarz, $$\left(1+\frac{2}{3}\right)\left({\sin }^{4}x+\frac{3}{2}{\cos }^{4}x\right)\ge ({\sin }^{2}x+{\cos }^{2}x{)}^2=1,$$so $${\sin }^{4}x+\frac{3}{2}{\cos }^{4}x\ge \frac{3}{5}.$$Equality occurs when $${\sin }^{4}x=\frac{9}{4}{\cos }^{4}x,$$or ${\tan }^{4}x=\frac{9}{4}.$ Thus, equality occurs for $x=\arctan \sqrt{\frac{3}{2}}.$ Hence, the minimum value is $\boxed{\frac{3}{5}}.$