Let y=f(x)≠0 satisfying 2(f(x))2−d2f(x)dx2f(x)+df(x)dx2=0&f(0)=f(1)=−1. Which of the following is/are CORRECT ?

2(f(x))2−f′′(x)f(x)+f′(x)2=0⇒f′′(x)f(x)−f′(x)2(f(x))2=2⇒ddxf′(x)f(x)=2⇒f′(x)f(x)=2x+a⇒df⁡(x)f(x)=(2x+a)dx⇒ln⁡|f(x)|=x2+ax+b⇒|f(x)|=ex2+ax+b Required area =∫01 (2x−1)ex2−xdx=−∫01/2 (2x−1)ex2−xdx+∫1/21 (2x−1)x2−xdxA=−ex2−x01/2+ex2−x1/21=2e1/4−1e1/4