Prove that the function f given by f x is increasing on I. . Prove that the function f given by f ( x ) = log sin x is increasing on , π and decreasing on π π , . .
Prove that the function f given by f ( x ) = log |cos x| is decreasing on , and increasing on , . Prove that the function given by f ( x ) = x – x + x – is increasing in R . . The interval in which y = x e –x is increasing is (A) (– ∞ , ∞ ) (B) (– , ) (C) ( , ∞ ) (D) ( , ) .
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 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. (i) 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 metres high, travels along a path given by h x , where x is the horizontal distance from the building and h ( x ) is the height of the ball .
What is the maximum height the ball will reach? (iii) An Apache helicopter of enemy is