Can you find …

… functions \(f(x)\) and \(g(x)\) so that \(f(g(x))\) has stationary points when \(x=-1\) and \(x=5\)?

… a function \(g(x)\) so that \(\ln{g(x)}\) has a local minimum?

… functions \(f(x)\) and \(g(x)\) so that \(f(g(x))\) has no stationary points?

The derivative of \(f(g(x))\) is the product of \(f'(g(x))\) and \(g'(x)\), provided these are defined.

When is a product zero?

When is a product positive or negative?

As well as making sure that \(f(g(x))\) is defined, we need to make sure that \(f'(g(x))\) and \(g'(x)\) are defined, so we need to keep an eye on the domains, ranges and derivatives of \(f(x)\) and \(g(x).\)

Here are a few more suggestions for separate parts of the problem.