<s>[INST] Maximum $\int_{0}^{y}\sqrt{x^4+\left(y-y^2\right)^2}dx$ where $y\in \left[0,1\right]$? How find maximum this integral $$\int_{0}^{y}\sqrt{x^4+\left(y-y^2\right)^2}dx$$ where $y\in \left[0,1\right]$?
 [/INST] $$\begin{align}\frac{d}{dy}(f(y)) &= \frac{d}{dy} \Bigg(\int_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx\Bigg) \\&= \int_{0}^{y}\frac{(y -y^2)(1-2y)}{\sqrt{x^4+(y-y^2)^2}} dx + \sqrt{y^4+(y-y^2)^2} . 1\end{align}$$
Now $$\begin{align}&\int_{0}^{y}\frac{(y -y^2)(1-2y)}{\sqrt{x^4+(y-y^2)^2}} dx + \sqrt{y^4+(y-y^2)^2} = 0 \\&\Rightarrow \int_{0}^{y}\frac{(y -y^2)(1-2y)}{\sqrt{x^4+(y-y^2)^2}} dx = - \sqrt{y^4+(y-y^2)^2} \\&\Rightarrow \int_{0}^{y}\frac{1}{\sqrt{x^4+(y-y^2)^2}} dx = \frac{\sqrt{y^4+(y-y^2)^2}}{(y-y^2)(2y-1)} \end{align}$$
Check the sign of $\frac{d}{dy}f(y)$ to see if the function $f(y)$ is decreasing, increasing.
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