By Cauchy-Schwarz's inequality:

\(L.H.S=\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}\)

\(=\dfrac{a^4}{ab+ac}+\dfrac{b^4}{bc+ab}+\dfrac{c^4}{ac+bc}\)

\(\ge\dfrac{\left(a^2+b^2+c^2\right)^2}{2\left(ab+bc+ca\right)}\ge\dfrac{\left(a^2+b^2+c^2\right)\left(ab+bc+ca\right)}{2\left(ab+bc+ca\right)}\)

\(\ge\dfrac{a^2+b^2+c^2}{2}=\dfrac{1}{2}=R.H.S\)

DONE!

L . H . S = a 3 b + c+ b 3 c + a+ c 3 a + bL.H.S=a3b+c+b3c+a+c3a+b

= a 4 a b + a c+ b 4 b c + a b+ c 4 a c + b c=a4ab+ac+b4bc+ab+c4ac+bc

≥ ( a 2 + b 2 + c 2 ) 2 2 ( a b + b c + c a )≥ ( a 2 + b 2 + c 2 ) ( a b + b c + c a ) 2 ( a b + b c + c a )≥(a2+b2+c2)22(ab+bc+ca)≥(a2+b2+c2)(ab+bc+ca)2(ab+bc+ca)

≥ a 2 + b 2 + c 2 2= 1 2= R . H . S

By Cauchy-Schwarz's inequality:

L.H.S=a3b+c+b3c+a+c3a+bL.H.S=a3b+c+b3c+a+c3a+b

=a4ab+ac+b4bc+ab+c4ac+bc=a4ab+ac+b4bc+ab+c4ac+bc

≥(a2+b2+c2)22(ab+bc+ca)≥(a2+b2+c2)(ab+bc+ca)2(ab+bc+ca)≥(a2+b2+c2)22(ab+bc+ca)≥(a2+b2+c2)(ab+bc+ca)2(ab+bc+ca)

≥a2+b2+c22=12=R.H.S