Show that the gradient of the curve \(y=x(x-3)^2\) is zero at the point \(P(1,4)\), and sketch the curve.

When curve sketching, good questions to ask include

  • What are the stationary points?

  • Where does the curve intercept the axes?

  • Are there any asymptotes?

  • How does the curve behave for large \(x\)?

The tangent at \(P\) cuts the curve again at \(Q\).

What is the equation of the tangent?

How does this help us to find \(Q\)?

Calculate the area between the chord \(PQ\) and the curve.

Can we see on our sketch the area we want to find? What techniques do we have for finding areas?