Problem

The equation of a curve is $y=f(x)$, where $f(x)=\frac{1}{2}x^{\frac{2}{3}}(x-2)^2$. The following points lie on the curve. Non-exact values of the $y$-coordinates are given correct to $6$ decimal places.

$$A(8,72),\quad B(8.001,k),\quad C(8.01,72.300388),\quad D(8.1,75.038882)$$

(a) Find the value of $k$. Give your answer correct to $6$ decimal places.

The table below shows the gradients of the chords $AB$ and $AC$, given correct to $4$ decimal places.

Chord	$AB$	$AC$	$AD$
Gradient of chord	$30.0039$	$30.0388$

(b) Find the gradient of the chord $AD$. Give your answer correct to $4$ decimal places.

(c) State what the values in the table suggest about the value of $f'(8)$.