Answer
See details below.
Work Step by Step
One can see that the function $ f(x)=\frac{x^2}{x^7+1} $ is continuous on $[0,1]$ and $ f(0)=0\neq f(1)=\frac{1}{2}$ and $0.4$ is between $ f(0)$ and $ f(1)$, then by the Intermediate Value Theorem the function $ f(x)$ takes on the value $0.4$ for some $ x\in (0,1)$.
