10 Prove that the logarithmic function is increasing on 0 11 Prove that the function fgiven by f x x x 1 is neither strictly increasing nor decreasing on 1 1 TU 12 Which of the following functions are decreasing on 0 2 A cos x B cos 2x C cos 3x D tan x 13 On which of the following intervals is the function fgiven by f x x 00 sin x 1 decreasing A 0 1 D None of these 14 For what values of a the function f given by f x x ax 1 ncreasing on 1 2 15 Let I be any interval disjoint from 1 1 Prove that the function f given by 1 f x x is increasing on I X 16 Prove that the function fgiven by f x log TC TU B TC C 0 2 2 decreasing on 160 increasing on TC T TU 2 37 3 2 a Ge MATHEMATICS 17 Prove that the function fgiven by f x log cos x is decreasing on 0 0 7 ar is increasing on thenERPo 0 1 18 Prove that the function given by f x x 3x 3x 100 is increasing in R 19 The interval in which y x e is increasing is A B 2 0 C 2 Rationalised 2023 24 and D 0 2 6 4 Maxima and Minima In this section we will use the concept of derivatives to calculate the maximum or minimum values of various functions In fact we will find the turning points of the graph of a function and thus find points at which the graph reaches its highest or and lowest locally The knowledge of such points is very useful in sketching the graph of a given function Further we will also find the absolute maximum and absolute minimum of a function that are necessary for the solution of many applied problems Let us consider the following problems that arise in day to day life 1 The profit from a grove of orange trees is given by P x ax bx where a b are constants and x is the number of orange trees per acre How many trees per acre will maximise the profit ii A ball thrown into the air from a building 60 metres high travels along a path