For the boundary value problem, $\,\,y''+\lambda y=0; y(0)=0,y(1)=0, \,\,\exists$ an eigenvalue $\lambda$ for which there corresponds an eigenfunction in $(0,1)$ that
- does not change sign
- changes sign
- is positive
- is negative
We have result says that eigen function corresponding to least eigen value does not change it sign and corresponding to $\lambda _k$ where $k>1$, eigen function change it sign hence option 1 and 2 are correct. Now here $y_1(x)= \sin(\pi x)$ which is positive and we know that if $y_1(x)$ is a solution then $-y_1(x)$ is also a solution. Then 3 and 4 are correct. Is the thought correct?