x2e2−∣x∣−1=0 f(x)=exe2x2−1∀x≥0 f′(x)=e2e2x{2xex−x2ex}=exe2⋅x(2−x) ⇒f increases in (0,2),f decreases in (2,∞)
Also f(0)<0&f(2)>0⇒ Exactly one root in (0,2) x→∞Limf(x)<0 ⇒ exactly one root in (2,∞) ⇒ exactly 2 roots in (0,∞) ⇒ equation has exactly 4 roots ∵f(x) is even function.