How do I solve this: Find the equation of the line joining the stationary points of y=x2(x−3) and the distance between them.?

Clarks

1 year ago

Find the equation of the line joining the stationary points of
y = x²(x − 3)
and the distance between them.
Step 1: What are stationary points?
Stationary points are special points on a graph where the slope (how steep the line is) becomes flat. This means the graph is not going up or down at those points.
To find them, we need to differentiate the equation — that's a fancy way to find the slope of the graph.

Step 2: First, expand the equation
We start with:
y = x²(x − 3)

Multiply it out:
y = x² × x − x² × 3
y = x³ − 3x²

Step 3: Find the derivative (slope)

Now, we find the derivative (this tells us the slope).
We call it dy/dx.

dy/dx = 3x² − 6x

Step 4: Set the derivative to 0 to find stationary points

3x² − 6x = 0

Factor it:
3x(x − 2) = 0

This gives two answers:
x = 0 and x = 2

Step 5: Find the y-values for these x-values

Put x = 0 and x = 2 back into the original equation:
y = x²(x − 3)

For x = 0:
y = 0²(0 − 3) = 0

For x = 2:
y = 2²(2 − 3) = 4 × (−1) = −4

So the two stationary points are:

(0, 0)

(2, −4)

Step 6: Find the equation of the line between them

Now we find the equation of the line that goes through these two points.

Use the formula for slope (m):
m = (y₂ − y₁) / (x₂ − x₁)
m = (−4 − 0) / (2 − 0) = −4 / 2 = −2

Now use the formula for a straight line:
y − y₁ = m(x − x₁)

Pick point (0, 0):
y − 0 = −2(x − 0)
So the equation is: y = −2x

Step 7: Find the distance between the two points

Use the distance formula:
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]

Plug in the values:
= √[(2 − 0)² + (−4 − 0)²]
= √[4 + 16] = √20 = 2√5

Final Answers:

Equation of the line: y = −2x

Distance between points: 2√5