10. [5 marks] Let f(x) = x2 - 12x + 11 on the interval [0, 10]. Find the values a and b such 11. In Assignment 3 you considered the swish and softplus functions, differentiable approxi- that f(x) attains its minimum at x = a and its maximum at x = b. mations to the rectifier function used in neural networks. An approximation that improves on these functions is the mish function f (x) = (2108 (1tex ) + 1) (a) [2 marks] Find the intercepts of f (x). Answer: (b) [3 marks] What polynomial best approximates f(x) for large negative values of a? Answer: Answer: Student number: 12 Student number: 13(c) [3 marks] What polynomial best approximates f(z) for large positive values of I? 12. Fechner's law describes the relationship between \"stimulus intensity\" I, such as the brightness of a light in lumens, and \"perceived intensity\" 11(1), such as the perceived brightness of a light by a viewer. It is given by = 1 ( ) . MI) k 0g m . where k and m are positive Constants. The domain is restricted to ac 2 m. (a) [3 marks] Find p'(ar), and state all intervals in the domain where p(m) is increasing. (Answers should be Written in the box. Justication should be outside the box.) (d) [3 marks] The degree 4 Taylor polynomial of f (z) at z = 0 is z+ ix lava 14 5 25 125 1575 ' What is f\"(0)? (b) [3 marks] Find p\"(z), and state all intervals in the domain where 11(1) is concave up. (Answers should be written in the box. Justication should be outside the box.) Student number: Student number: (c) [3 marks] Draw a large sketch of the graph of 17(1) on the axes below, clearly (e) [2 marks] Stevens' power law is sometimes taken by psychophysicists (who study labelling any intercepts, asymptotes, extrema and inection points. stimuli and perceptions) to supersede Fechner's law. It proposes that perceived intensity 1M1) is related to stimulus intensity 1' according to y WI) = kzi where a is a parameter that depends on the type of stimulus, and k is the same positive constant as in Fechner's law. If p(z) and 1M1) are meant to describe the same relationship, in which of the following intervals is the parameter (17 Remember to justify your answer. (d) [3 marks] At a given stimulus intensity I, the dzscn'mmatum threshold is the mini, mum increase in stimulus intensity such that the corresponding increase in perceived intensity is equal to or greater than a constant Ap. According to this model, are discrimination thresholds at high stimulus intensities greater than discrimination thresholds at low stimulus intensities? Give your answer and justify it in a few sentences. Student number: Student number