2021 May TZ1 P1 Q8

Let \(y = \frac{\ln{x}}{x^4}\) for \(x > 0\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1 - 4\ln{x}}{x^5}\).

   [3]

Consider the function defined by \(f(x) = \frac{\ln{x}}{x^4}\) for \(x > 0\) and its graph \(y = f(x)\).

2. The graph of \(f\) has a horizontal tangent at point \(\mathrm{P}\). Find the coordinates of \(\mathrm{P}\).

   [5]

3. Given that \(f''(x) = \frac{20\ln{x} - 9}{x^6}\), show that \(\mathrm{P}\) is a local maximum point.

   [3]

4. Solve \(f(x) > 0\) for \(x > 0\).

   [2]

5. Sketch the graph of \(f\), showing clearly the value of the \(x\)-intercept and the approximate position of point \(\mathrm{P}\).

   [3]