Let f be continuous on [a,b] and differential on (a,b). Prove that if a >= 0 there are x1, x2, x3 ∈ (a,b) such that

${f}^{\prime}({x}_{1})=(b+a)\frac{{f}^{\prime}({x}_{2})}{2{x}_{2}}=({b}^{2}+ba+{a}^{2})\frac{{f}^{\prime}({x}_{3})}{3{x}_{3}^{2}}$

I think this problem will use the generalized mean value theorem to solve. However, I don't know how to apply it. Can you suggest a solution? Thank you very much!

${f}^{\prime}({x}_{1})=(b+a)\frac{{f}^{\prime}({x}_{2})}{2{x}_{2}}=({b}^{2}+ba+{a}^{2})\frac{{f}^{\prime}({x}_{3})}{3{x}_{3}^{2}}$

I think this problem will use the generalized mean value theorem to solve. However, I don't know how to apply it. Can you suggest a solution? Thank you very much!