<s>[INST] Find the minimum value of $f$ on $\{(a,b,c)\in \mathbb{R^3}\mid a+b+c=1\}$ where $f(a,b,c)=\int_{0}^1 (a+bx+cx^2)^2dx$. Find the minimum value of $f$ on $\{(a,b,c)\in \mathbb{R^3}\mid a+b+c=1\}$ where $f(a,b,c)=\int_0^1 (a+bx+cx^2)^2 \, dx$.
What I did is 
$$
\int_0^1 (a+bx+cx^2)^2 \, dx=a^2+a \left( b+\frac{2c}{3} \right) + \frac{b^2}{3} + \frac{bc}{2}+\frac{c^2}{5}.
$$
Then we consider the function $F(a,b,c)=a^2+a(b+\frac{2c}{3})+\frac{b^2}{3} + \frac{bc}{2} + \frac{c^2}{5}-\lambda(a+b+c-1)$. Thus, we can consider $\nabla F=0$ with $a+b+c=1$ and try to find $a,b$ and $c$.
However, since the function and the constraint seem to be related, I am curious whether there is some other way to see the minimum without computing Lagrange multiplier. Thank you.
 [/INST] Just substitute $a=1-b-c$ and minimize the function of two variables
$$g(b,c)=f(1-b-c,b,c)=\int_0^1(1-b-c+bx+cx^2)^2 dx.$$
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