$j[y]=\int_{0}^{1}[(y')^{2}-(y')^{4}]dx$ ,subject to condition $y(0)=0,y(1)=0.$A broken extremal is a continuous extremal whose derivative has jump discontinuities at a finite number of points.Then whichof the following is /are true?
- 1).There are no broken extremals and $y=0$ is an extremal.
- 2).There is a unique broken extremal.
- 3).There exist more then one and finitely many broken extremals.
- 4).There exist infinitely many broken extremals.
Here the Extremal function will be $y=ax+b$ type but if we apply the boundery conditions then we get $y=0$ , answer should be' $1$' option but answer is '$3$' ,i am not getting the right answer please help
thank you.