Understand and use the second derivative test for finding local maxima and minima.

The number of visitors to a museum during opening hours is modelled by the function \( V(t) = 85t - 6t^2 + 45 \), where \( t \) is time in hours after opening and \( 0 \leq t \leq 10 \). (a) Find \( V'(t) \) and hence determine the value of \( t \) when \( V'(t) = 0 \). (1 mark) (b) Find \( V''(t) \). (1 mark) (c) Use your result from part (b) to explain how the value of the second derivative is consistent with the number of visitors being a maximum. (1 mark)

A function \( f(x) \) is continuous and differentiable for all real \( x \). The first derivative is \( f'(x) = 3x^2 - 12x + 9 \). The nature of the stationary points of \( f(x) \) is best described as

A coastal town models the daily number of tourists, N(t), in hundreds, where t is the number of weeks after January 1st, and 0 ≤ t ≤ 26. The rate of change of the number of tourists is given by N'(t) = 2t² - 20t + 32. (a) Determine the value(s) of t at which N'(t) = 0. [1 mark] (b) Calculate N''(t) and evaluate N''(t) at each value found in part (a). [2 marks] (c) Use your results from part (b) to explain how the value(s) of the second derivative are consistent with the number of tourists being at a local maximum or local minimum. [2 marks]