Question # Find the dimensions of the rectangle of largest area that

Negative numbers and coordinate plane
ANSWERED Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. (Round your answers to the nearest hundredth.)
$$\displaystyle{y}={6}-{x}^{{2}}$$ 2021-08-15
Area of the rectangle is A=xy Substitue $$\displaystyle{y}={6}-{x}^{{2}}$$ in the formula of the area. $$\displaystyle{A}={x}{\left({6}-{x}^{{2}}\right)}$$
$$\displaystyle={6}{x}-{x}^{{3}}$$ Diferentiate area with respect to x and equate to 0. $$\displaystyle{A}'={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({6}{x}-{x}^{{3}}\right)}$$
$$\displaystyle={6}-{3}{x}^{{2}}$$
$$\displaystyle{6}-{3}{x}^{{2}}={0}$$
$$\displaystyle{3}{x}^{{2}}={6}$$
$$\displaystyle{x}=\pm\sqrt{{{2}}}$$
Substitute $$\displaystyle{x}=\pm\sqrt{{{2}}}\ \in\ {y}={6}-{x}^{{2}}$$ to get value of height, $$\displaystyle{y}={6}-{x}^{{2}}$$
$$\displaystyle={6}-{\left(\sqrt{{{2}}}\right)}^{{2}}$$
$$\displaystyle={4}$$ Hence. the dimensions of rectangle are, $$\displaystyle{2}\sqrt{{{2}}}\times{4}.$$