## Exam-Style Question on Differentiation## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 561. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is not allowed.

The function \(f\) is such that \(f(x) = \frac{\ln2x}{x^3} \) where \(x \gt 0\).

(a) Find the first derivative of the function, \( f'(x) \).

(b) The graph of \( y=f(x) \) has a horizontal tangent at the point T. Find the coordinates of T.

(c) Show that T is a local maximum point by considering the second derivative, \( f''(x) \).

(d) Find the values of \(x\) for which \( f(x) \gt 0 \).

(e) Sketch the graph of \( f \) showing clearly the value of the x-intercept and the approximate position of point T.

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