1. Show that the curve \(y=x(5-x)^4\) has a turning point where \(x=5\). Determine whether this turning point is a maximum or a minimum.

How can we find the turning points of a function?

What methods do we know for determining the nature of a turning point?

Which method works best in this case?

  1. It is given that that \(\dfrac{dy}{dx}\) is inversely proportional to \(x^2\) and that \(y\) and \(\dfrac{dy}{dx}\) are each equal to \(1\) when \(x=2\). Express \(y\) in terms of \(x\).

What does it mean for \(a\) to be inversely proportional to \(b\)? Can you write this mathematically?

Can we now replace \(a\) with \(\dfrac{dy}{dx}\), and \(b\) with \(x^2\), and use the information in the question?