Exploring the Slope of the Curve x3 - 2xy 0

The curve described by the equation x3 - 2xy 0 has been a topic of interest in calculus, particularly in the analysis of slopes at various points. This article aims to explore the point at which this curve has a slope of 3.

Deriving the Slope of the Curve

To determine the slope of the curve at any point, we start by isolating y:

x3 - 2xy 0

Rearranging the equation to solve for y:

y x2

Next, we differentiate y with respect to x to find the slope:

(frac{dy}{dx} 2x)

Setting the Slope to 3

We want to find the points where the slope is 3. Setting the slope equation equal to 3:

2x 3

However, solving for x yields:

(x frac{3}{2})

Substituting (x frac{3}{2}) back into the equation (y x^2):

(y left(frac{3}{2}right)^2 frac{9}{4})

But this solution does not satisfy the original curve equation, indicating that there is no point on the curve where the slope is 3.

Further Analysis of the Curve

To gain a deeper understanding, we can examine the extrema of the curve. First, we find the derivative and set it to 0:

(frac{dy}{dx} 2 - 3x^2 0)

Solving for x at the extrema:

(x pmsqrt{frac{2}{3}})

For x values outside this range, the curve is decreasing and has a negative slope. Between these values, the curve increases and has a positive slope. The maximum positive slope occurs at x 0:

(y' 2)

Graphical Interpretation

The graph of the curve x3 - 2xy 0 visually confirms that the slope never reaches 3. The curve has a maximum positive slope of 2 at the point where x 0.

The exploration of the slope of the curve demonstrates the importance of calculus in understanding the behavior of functions. While no point on the curve has a slope of 3, the curve does exhibit interesting properties at its extrema and inflection points.