Mathematics - Application of Derivative Question with Solution | TestHub

If the sum of two positive quantities is a constant, then their product is maximum, when they are equal. $\therefore {\mathrm{a}}^2{\mathrm{x}}^4\cdot {\mathrm{b}}^2{\mathrm{y}}^2$ is maximum when $$a^2{x}^4=b^2{y}^4=\frac{1}{2}(a^2{x}^4=\frac{1}{2}\left(a^2{x}^4+b^2{y}^4\right)=\frac{c^4}{2}$$ $\therefore$ maximum value of $a^2{x}^4\cdot b^2{y}^4=\frac{c^4}{2}\cdot \frac{c^4}{2}=\frac{c^8}{4}$ Maximum value of $xy={\left(\frac{c^8}{4a^2{b}^2}\right)}^{1/4}=\frac{c^2}{\sqrt{2ab}}$