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## Based on the differentiation…

Based on the differentiation $dA / db$, It can be seen that $a$ is constant from the equation.

$$A = \frac{1}{4}(b + a) \sqrt{4a^2 - (b - a)^2}$$

Where $u = b + a$ and $v = \sqrt{4a^2 - (b - a)^2}$ from which $\dfrac{du}{db} = 1$. For $dv$ however, we will use the formula $d\left( \sqrt{u} \right) = \dfrac{du}{2\sqrt{u}}$. Hence, $\dfrac{dv}{db} = \dfrac{-2(b - a)}{2\sqrt{4a^2 - (b - a)^2}}$

Now, apply the whole $d(uv)$ to the equation:

$$\dfrac{dA}{db} = \dfrac{1}{4} \left[ (b + a) \cdot \dfrac{-2(b - a)}{2\sqrt{4a^2 - (b - a)^2}} + \sqrt{4a^2 - (b - a)^2} \cdot 1 \right] = 0$$

The zero is the concept of maxima and minima. You need to go back to the basic concept of optimization to understand why the equation is equated to zero.