<s>[INST] Find the minimum value of the ratio $\frac{S_1}{S_2}$ using given data 
3 points $O(0, 0) , P(a, a2 ) , Q(b, b2 )$ are on the parabola $y=x^2$. Let S1 be the area bounded by the line PQ and the parabola and let S2 be the area of the triangle OPQ, then find min of $\frac{S1}{S2}$

$$S_1 = \frac 12 (a+b)(a^2+b^2) -\int_{-b}^a x^2.dx$$
$$=\frac 16 (a^3+b^3) +\frac 12 (ab^2+a^2b)$$
And $$S_2=\frac 12 (a^2b+ab^2)$$
How do I solve from here?
 [/INST] $$
\dfrac{S_1}{S_2}=\frac{\frac{1}{2} \left(a^2 b+a b^2\right)+\frac{1}{6} \left(a^3+b^3\right)}{\frac{1}{2} \left(a^2 b+a b^2\right)}=\frac{(a+b)^2}{3 a b}\geq \dfrac{4ab}{3ab}=\dfrac{4}{3}
$$
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