Use the derivatives of the function to sketch the graph y = 2xsqrt(6 − x^2).?

y = 2x √(6 - x²) => dy/dx = 2 √(6 - x²) - 2x² / √(6 - x²)

=> dy/dx = [2 (6 - x²) - 2x²] / √(6 - x²)

Looking first at y, we see that it's zero when x = +/- √6, so mark the points (√6, 0) and (-√6, 0) on the coordinate plane. Also, x = 0 => y = 0.

Notice that x must always lie between these values, or y has no real value.

Also, easily, x = +/- 1 => y = +/- 2√ 5 and x = +/- 2 => y = +/- 4√ 2. You may as well mark all these points too, so you now have SEVEN points to help you get a good idea of what the graph looks like.

You can see that it's an "odd" function, i.e. y(-x) = -y(x), i.e. it's got rotational symmetry order 2 about the origin. So once you've got the positive bit you immediately know the negative bit.

Now looking at dy/dx, this is zero when 12 - 4x² = 0, i.e. x = +/- √3, so that tells you where the local minimum (for x = √3) and maximum (for x = -√3) are. Of course, the y values for these will be +/- 4√3.

Finally, you can see that at x = +/- √6, dy/dx becomes infinite, so you should make the graph 'vertical' as it approaches these points.