32 4 1 1 4 0 as we move from left to right the height of the graph decreases 2 1 X J X f x height of graph at x O PINN Fig 6 1 First consider the graph Fig 6 1 to the right of the origin Observe that as we move from left to right along the graph the height of the graph continuously increases For this reason the function is said to be increasing for the real numbers x 0 Strictly Increasing function i 1 Now consider the graph to the left of the origin and observe here that as we move from left to right along the graph the height of the graph continuously decreases Consequently the function is said to be decreasing for the real numbers x 0 Rationalised 2023 24 X X 2 We shall now give the following analytical definitions for a function which is increasing or decreasing on an interval 3 Definition 1 Let I be an interval contained in the domain of a real valued function f Then f is said to be i increasing on I if x x in 1 f x x for all x x I ii decreasing on I if x x in I x x for all x x I iii constant on I if f x c for all x I where c is a constant 2 Fig 6 2 2 4 as we move from left to right the height of the graph increases iv decreasing on I if x x in I f x x for all x x I v strictly decreasing on I if x x in 1 x x for all x x I For graphical representation of such functions see Fig 6 2 Strictly Decreasing function ii 4 1 9 4 APPLICATION OF DERIVATIVES X X O 153 ist X Neither Increasing nor Decreasing function 20 We shall now define when a function is increasing or decreasing at a point Definition 2 Let x be a point in the domain of definition of a real valued function f Then f is said to be increasing decreasing at x if there exists an open interval I containing x such that fis increasing decreasing respectively in I Let us clarify this definition for the case of increasing function Example 7 Show that the function given by f x 7x 3 is increasing on R