Have you tried sketching the functions we have been given?

If we want…

If we want a curve given by \(y=f(x)\) to have a stationary point at \((a,b)\) where \(a\) and \(b\) are real numbers, we want \(f'(a)=0\) and \(f(a)=b\).

How many stationary points can a quadratic or cubic have?

What about a degree \(4\) or \(5\) polynomial?

If we want a curve to have a local minimum or local maximum, what could the curve “look like”?

If \(g(x)\) is a function with a local maximum at \(x=2\), can we use this information to create a function that has a local minimum at \(x=2\)?