Understand the concepts of concavity and points of inflection and their relationship with the second derivative.

The second derivative of a function is given by $f''(x) = 3x^2 - 12$. On which interval is the graph of $f(x)$ concave down?

Consider the function \( g(x) = \frac{x^3}{6} - \frac{5x^2}{4} + 3x + 1 \). (a) Determine \( g''(x) \). (1 mark) (b) Find all \( x \)-values where the concavity of \( g(x) \) changes. (1 mark) (c) Establish the intervals on which \( g(x) \) is concave up and concave down. (2 marks) (d) Hence, state the coordinates of any points of inflection on the graph of \( y = g(x) \). Give your answer correct to two decimal places. (1 mark)

Consider the function \( g(x) = \frac{x^3}{3} - 2x^2 + 3x + 5 \). (a) Determine the second derivative \( g''(x) \). (1 mark) (b) Find the \( x \)-coordinate where \( g''(x) = 0 \). (1 mark) (c) Use the second derivative to demonstrate that the function has a point of inflection at this \( x \)-value. (2 marks)