at r = cm is (A) π (B) π (C) π (D) π . The total revenue in Rupees received from the sale of x units of a product is given by R( x ) = x + x + . The marginal revenue, when x = is (A) (B) (C) (D) . Increasing and Decreasing Functions In this section, we will use differentiation to find out whether a function is increasing or decreasing or none.
Consider the function f given by f ( x ) = x , x ∈ R . The graph of this function is a parabola as given in Fig . . Fig .
First consider the graph (Fig . ) 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 > .
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 < . We shall now give the following analytical definitions for a function which is increasing or decreasing on an interval. Definition 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 I ⇒ f ( x ) < f ( x ) for all x , x ∈ I. (ii) decreasing on I, if x , x in I ⇒ f ( x ) < f ( x ) for all x , x ∈ I. (iii) constant on I, if f(x) = c for all x ∈ I, where c is a constant. f ( x ) = x