– – Values left to origin as we move from left to right, the height of the graph decreases f ( x ) = x Values right to origin as we move from left to right, the height of the graph increases (iv) decreasing on I if x < x in I ⇒ f ( x ) ≥ f ( x ) for all x , x ∈ I. (v) strictly decreasing on I if x < x in I ⇒ f ( x ) > f ( x ) for all x , x ∈ I. For graphical representation of such functions see Fig . .
Fig . We shall now define when a function is increasing or decreasing at a point. Definition 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 f is increasing, decreasing, respectively, in I.
Let us clarify this definition for the case of increasing function. Example Show that the function given by f ( x ) = x – is increasing on R . Solution Let x and x be any two numbers in R . Then x < x ⇒ x < x ⇒ x – < x – ⇒ f ( x ) < f ( x ) Thus, by Definition , it follows that f is strictly increasing on R .
We shall now give the first derivative test for increasing and decreasing functions. The proof of this test requires the Mean Value Theorem studied in Chapter . Theorem Let f be continuous on [ a, b ] and differentiable on the open interval ( a,b ). Then (a) f is increasing in [ a,b ] if f ′ ( x )