Calcola la lunghezza di una circonferenza sapendo che il suo raggio è congruente al lato di un triangolo equilatero avente il perimetro di 54 cm . \( \quad[113,04 \mathrm{~cm}] \)

Let \( f(x)=\frac{\mu}{\ln x} \) for \( x>1 \) (a) Derive expressions for \( f^{\prime}(x) \) and \( f^{\prime \prime}(x) \), simplifying your answers. (b) Obtain the coordinates and nature of the stationary point of the curve \( y=f(x) \). (c) Obtain the coordinates of the point of inflexion.

24 Consider the equation of the second degree in \( x: x^{2}-(m+3) x+m=0 \), where \( m \) is a real parameter. Show that there is between the roots, when they exist, an independent relation from \( m \). (Eliminate \( m \) between \( S \) and \( P) \).

c. \( y=7.5 x+15.5 \)

Evaluate \( \sin \left(\tan ^{-1}\left(-\frac{7}{10}\right)\right) \), giving an exact answer.

Define the term 'convex polygon' and provide an example.