Increasing and Decreasing Functions

Increasing and decreasing functions

This is my third post in the series of “Applications of derivatives”. The previous two were based on “Tangent and Normal” and “Maxima and Minima”.In this post, we shall learn about increasing and decreasing functions. That is one more application of derivatives.

Increasing and Decreasing Functions- We shall first learn about increasing functions

Increasing Function-

(a) Strictly increasing function- A function f (x) is said to be a strictly increasing function on (a, b) if x1< x2  \Rightarrow f(x1) < f (x2) for all xl, x2 \in (a, b).Thus, f(x) is strictly increasing on (a, b) if the values of f(x) increase with the increase in the values of x.Refer to the graph in below-given figure  \Downarrow <img src="increasing decreasing function.jpg" alt="increasing decreasing function">

Graphically, f (x) is increasing on (a, b) if the graph y = f (x) moves up as x moves to the right. The graph in Fig.1  is the graph of a strictly increasing function on (a, b). 

Classification of Strictly Increasing Functions on the basis of Shape-

(i) Concave up– When f'(x)>0 and f”(x)>0 for all x  \in  domain

Refer to the graph in below-given figure  \Downarrow

<img src="increasing decreasing function.jpg" alt="increasing decreasing function">

(ii) When f'(x)>0 and f”(x)=0 for all x  \in  domain

Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

 

(iii) Concave down or convex up – When f'(x)>0 and f”(x)<0 for all x  \in  domain

Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

(b) Only Increasing or non-decreasing functions- A function is said to be non-decreasing if for  {x_1} > {x_2}  \Rightarrow  f({x_1}) \ge f({x_2}) as shown in the graph, for AB and CD;

{x_1} > {x_2}     \Rightarrow   f({x_2}) > f({x_1})

and for the portion BC ;  {x_2} > {x_1}      \Rightarrow f({x_2}) = f({x_1})

thus overall we can say that  {x_2} > {x_1}    \Rightarrow f({x_2}) \ge f({x_1}) so it is obvious for increasing or non-decreasing functions, f'(x)  \ge 0 with equality holding in interval like BC.

Refer to the graph in below-given figure  \Downarrow <img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

 

Decreasing Functions-

(a) Strictly Decreasing Function-A function f (x) is said to be a strictly increasing function  on (a, b) if x1< x2    \Rightarrow f (x1) > f (x2) for all xl, x2 \in (a, b)

Thus, f (x) is strictly decreasing on (a, b) if the values of f (x)  decrease with the increase in the values of x.

Graphically, f (x) is a decreasing function on  (a, b) if the graph y = f (x) moves down as x moves to the right. The graph in Fig. is the graph of a strictly decreasing function on      (a, b). Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

Classification of Strictly Decreasing Functions on the basis of Shape-

(i) Concave up or Convex down-When f'(x)<0 and f”(x)>0 for all x  \in  domain

Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

(ii) When f'(x)<0 and f”(x)=0 for all x  \in  domain

Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

(iii) Concave down or convex up – When f'(x)<0 and f”(x)<0 for all x  \in  domain

Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

(b) Only decreasing or non-increasing functions- A function is said to be non-increasing if for  {x_2} > {x_1}     \Rightarrow   f({x_1}) \ge f({x_2}) as shown in the graph, for AB and CD ;

{x_2} > {x_1}     \Rightarrow   {f({x_2}) < f({x_1})}

and for the portion BC ;  {x_2} > {x_1}      \Rightarrow f({x_2}) = f({x_1})

thus overall we can say that  {x_2} > {x_1}     \Rightarrow f({x_2}) \ge f({x_1})

Obviously for this  f'(x) \le 0 where equality holds for horizontal part of the graph i.e, in the interval of BC.

Refer to the graph in below-given figure  \Downarrow

<img src="increasing and decreasing function.jpg" alt="increasing and decreasing function">

Monotonic Function: A function is said to be monotonic in the given interval (a,b) If it is either increasing or decreasing in the given interval.

Definition: A function is said to be increasing (decreasing) at a point x0 if there is an       interval (x0-h,x0+h) containing x0 such that f(x) is increasing(decreasing) on (x0-h,x0+h)

Definition: A function is said to be increasing or decreasing on [a,b] if it is increasing(decreasing) at (a,b) and it is also increasing (decreasing) at x=a and x=b

Conditions for Increasing and Decreasing Functions- We can easily identify  increasing and decreasing functions with the help of differentiation.

(i) If f'(x)>0 for all x \in  (a,b), then f(x) is increasing on (a,b)

(ii) If f'(x)<0 for all x \in  (a,b), then f(x) is decreasing on (a,b)

 

Properties of Monotonic Functions-

(a) If f(x) is a function that is strictly increasing in the interval [a,b] then inverse of given function (f-1) exists and  f-1  is also strictly increasing function

(b) If f(x) and g(x) are functions that are strictly increasing or decreasing in the interval [a,b] then composite of given functions gof(x) is  also strictly increasing or decreasing

(c) If f(x) increasing function then its reciprocal 1/f(x) is decreasing function

(d) If f(x) and g(x) are both increasing functions then (f+g)(x) is also increasing

(e) If f(x) is increasing and g(x) is decreasing or vice-versa then gof(x) is decreasing

Here I am posting a worksheet on Increasing and Decreasing Functions and another worksheet with 200 (approx.) questions on Applications of Derivatives.

 APPLICATION OF DERIVATIVES.pdf

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 Increasing & Decreasing Function (09-10-15).pdf

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