Derivative Optimization Problem

Question

I need help with finding the area of the largest rectangle in an ellipse from $y^2 + (x^2)/4 = 1$.

I got it to y = $\sqrt{ 1 - (x^2)/4}$ but then I don't really know what to do, please help.

Answer 1

I guess the area is defined as $x\times y$. So far, you got $y=\sqrt{1-x^2/4}$, therefore

$$A(x)=x\times\sqrt{1-x^2/4} \quad \text{for}\quad -2 \leq x \leq 2$$

Negative values do not make sense, so these might be ruled out. Also, the area would be $0$ for $x=2$ and $x=0$, so $0 < x < 2$.

Now you have to find the maximum of this function by differentiating.

$$\frac{dA}{dx}=-\frac{x^2-2}{2\sqrt{1-x^2/4}}$$

The slope at the highest point is $0$, so

$$0=-\frac{x^2-2}{2\sqrt{1-x^2/4}}$$

A fraction is only $0$, if the numerator is $0$:

$$0=x^2-2$$

Finally, $$x = \sqrt{2}$$

With this solution, you can find the final maximum area of the rectangle.