Determine the values of $x$ for which $f(x)$ is a decreasing function.

There is a point of inflexion, $P$, on the curve $y = f(x)$.

Find the coordinates of $P$.

attempt to differentiate $f(x) = {x^3} - 3{x^2} + 4$     M1

$f'(x) = 3{x^2} - 6x$     A1

$= 3x(x - 2)$

(Critical values occur at) $x = 0,{\text{ }}x = 2$     (A1)

so $f$ decreasing on $x \in ]0,{\text{ }}2[\;\;\;({\text{or }}0 < x < 2)$     A1

[4 marks]

$f''(x) = 6x - 6$     (A1)

setting $f''(x) = 0$     M1

$\Rightarrow x = 1$

coordinate is $(1,{\text{ }}2)$     A1

[3 marks]

Total [7 marks]