<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] In general, if you have a fraction involving $a,b$ where each term in the numerator and denominator has the same degree $d$, you can divide through by say $b^d$ to get an expression in $\frac{a}{b}$. Here, this would look like $$\frac{S_1}{S_2} =\frac{ \frac 16 (a^3+b^3) +\frac 12 (ab^2+a^2b)}{\frac 12 (a^2b+ab^2)}  =\frac{ \frac 16 ((a/b)^3 +1) +\frac 12 (a/b+a^2/b^2)}{\frac 12 (a^2/b^2+a/b)} $$ Let $a/b =t$, then $$\frac{S_1}{S_2} = \frac{ t^3/6 +1/6 + t/2 +t^2/2}{t^2/2 +t/2} $$ And then you can follow the usual optimization procedure.
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