Move the right-hand term to the other side of the equals sign:
Cross-multiply: 2(100 − x^2 ) = 2x^2
Distribute: 200 − 2x^2 = 2x^2
And solve: 200 = 4x^2
x = = 5 , so 2x = 10
Now we can solve for y: y =
Therefore, the dimensions are 10 by 10.
- Maximum at (0, 0); Minima at (2, −4) and (−2, −4); Points of inflection at and
.
First, we can easily see that the graph has x-intercepts at x = 0 and x = ± . Next, we take the
derivative: = x^2 −4x. Next, we set the derivative equal to zero to find the critical points.
There are three solutions: x = 0, x = 2, and x = −2.
We plug these values into the original equation to find the y-coordinates of the critical points:
When x = 0, y = −2(0)^2 = 0. When x = 2, y = −2(2)^2 = −4. When x = −2, y =
−2(−2)^2 = −4. Thus, we have critical points at (0, 0), (2, −4), and (−2, −4). Next, we take the
second derivative to find any points of inflection. The second derivative is = 3x^2 − 4,
which is equal to zero at x = and x = − . We plug these values into the original equation
to find the y-coordinates. When x = : . When x = −
