Inflection point

A curve's Inflection point is the point at which the curve's concavity changes. A point that separates the convex part of the curve from part of the curve is called a point of inflexion.

Consider the curve y=f(x) in [ a,b ]. Let it be continuous and possessing tangent at every point in ( a,b ).
Draw a tangent at any point P ( c,f(c) ) on the curve. Let us assume that this tangent is not parallel to Y-axis so that f'(c) is some finite number.

Point Of Inflection 

       The portion of the curve on the two side of P lies on different side of the tangent at P i.e, the curve crosses the tangent at P. In this case we say that P is a point of inflection  on the curve.
         So, at a point of inflection the curve change from concave upward so convex downward or vice-versa.

   So at a point of inflexion f''(x)=0.

☆Working Method Of Find The Point Of inflection 

(1). Evaluate 

(2). Find the value of x which satisfy  and also the value of x where   does not exist.
       Such values x=a,b,c,....(say) are the possible point of inflection.

(3). x=0 will be point of inflection
      If (1). Either   change sign at x=a
      or(2).  exist and is non-zero at x=a.

Note1.    is not a sufficient condition for graph of f to have a point of inflection.

Note2. If at a point, x=c,  when n is even, then x=c is not a point of inflection.

Note3. If at a point, x=c,  for some even n and  , then the curve has a point of inflection at x=c.

Example

Q1. Find the point of inflection on the graphs of the function  
Sol.
        

   
        
Now 
        

    x=0 is not a point of inflection.

Concavity and convexity

When you plot a function in the graph, if the curve opens towards the positive Y-axis then it is said to be concave up (or convex down) function while if it opens downwards then the curve is said to be concave down  (or convex up ) function.

Concavity and convexity

Consider the curve y=f(x) in [ a,b ]. Let it be continuous and possessing tangent at every point in ( a,b ).
Draw a tangent at any point P ( c,f(c) ) on the curve. Let us assume that this tangent is not parallel to Y-axis so that f'(c) is some finite number.

Now there are three mutually exclusive possibilities to consider.

1. Concave upward 

        A portion of the curve on both side of P, however small it may be, lies above the tangent at P ( i.e. towards the +ve direction of Y-axis ). In this case, we say that the curve is concave upward or convex downward at P. Such curve "hold water".
            As x-increase, f'(x) is either of the same sign and increasing or change sign from -ve to +ve. In either case, the slop f'(x) is increasing and f''(x) >0. Such graphs are bending upward or the portion lies above the chord.

2. Concave downward 

      A portion of the curve on both side of P, however small it may be, lies below the tangent at P ( i.e. towards the -ve direction of Y-axis ). In this case, we say that the curve is convex upward or concave downward at P. 
          As x-increase, f'(x) is either of the same sign and decreasing or change sign from +ve to -ve. In either case, the slop f'(x) is decreasing and f''(x) <0. Such graphs are bending downward or the portion lies below the chord.

3. Point of inflection 

       The portion of the curve on the two side of P lies on different side of the tangent at P i.e, the curve crosses the tangent at P. In this case we say that P is a point of inflection on the curve.
         So, at a point of inflection the curve change from concave upward so convex downward or vice-versa.

   So at a point of inflection f''(x)=0.

☆Definition of concavity and convexity

      A curve is said to be concave downward ( or convex upward ) on the interval ( a,b ) if all the points of the curve lies below any tangent to it on that interval. It is said to be concave upward ( or convex downwards ) on the interval ( a,b ) if all the points of the curve lies above any tangent to it on that interval.

☆Working method for concavity and convexity 

   (1). Evaluate 

   (2). Find the interval ( a,b ) for which .
     Then ( a,b ) is the interval of being convex downward.

   (3). Find the interval ( a,b ) for which .
     Then ( a,b ) is the interval of being convex upward.

Examples

    Q1. Prove that the curve  is concave upward for all x∈ℝ.
    Sol.
             Here  
                   
           and       

               
The curve   is concave upward 

   Q2. Prove that the curve  is everywhere concave downward for x >0.
   Sol.
            Here  , x >0
                     
                         
               Now 
Hence the curve is concave downward or convex upward for x >0.

Intervals

Interval is a way to describe continuous set of real number by the number that bound them. However, they are not meant to denote a specific point. Rather, they are meant to describe an inequality or system of inequalities.

Let a and b be two distinct real number with a<b then,

1. Open interval

The set of all real number between a and b said to form an open interval from a to b denoted by (a,b). In symbols

           
Geometrically the open interval (a,b) is represented on the real line as

For example= Inequality: -1<x<5
                            Interval    : (-1,5)
In this case, x does not equal -1 and 5. When both of the end point are excluded from the interval, the interval is open interval.

2. Closed interval 

The set of all real number between a and b including the end points and b is said to form a closed interval and is denoted by [a,b].
In symbols 
Geometrically, the closed interval [a,b] is represented on the real line as

For example= inequality: 3≤x≤9
                          interval    : [3,9]
In this case, x could equal 3 or 9 when both of the end point are included in the interval, the interval is a closed interval.

3. Half-open intervals

An interval in which one end point is included are the other end is excluded is called half-open interval.
In symbols, 
Geometrically, [a,b) is represented on the line as

For example= inequality: -3≤x<5
                          interval    : [-3,5]
In this case, x could equal -3 but it cannot equal 5. When one of the end points is included in the interval but the other is not, the the interval is a half-open interval.

Similarly 
Geometrically, (a,b] is represented on the real line as

The interval defined above are called finite intervals. Now we define infinite intervals.

Infinite interval. The set of all real number x such that x>a forms an infinite set and is denoted by (a,∞).
Symbolically  geometrically

If an interval has no lower bound or upper bound then the -∞ and ∞ symbol are used. These symbols are always used with a parentheses bracket, because infinity is not a number that can be included in a set

For example. inequality: x≤7
                          interval   : (-∞,7)
                     
                         inequality: x>3
                         interval    : (3,∞).

L'Hospital's Rule

L'Hospital's Rule is a method for finding the value of a function using derivatives. This method is use when, the value at a point don't exist.

Suppose there are continuous function f(x) and g(x) that are both zero at x=a. Then, the limit   cannot be found by substituting x=a since it give 0/0 which cannot be evaluated we use 0/0 as a notation for an expression known as an indeterminate form may be evaluated by cancellation or rearrangement of terms. However, this does not always work.
      For example, how would you evaluate   ? Obviously inserting x=0 will give an indeterminate form of 0/0 and, in this case you can neither use algebraic manipulation nor rearrangement of terms to reduce this expression into a form that yield a valid limit.

L'Hospital's Rule

Statement: If f and g are differentiable function such that

1.   or

     ;

2. f'(x), g'(x) both exist and
g'(x)≠0 ∀ x∈(a-δ,a+δ), δ>0 except possibly at x=a;

3.  exist
Then

Examples

Q1. Evaluate the limit : .
Ans.
       Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since sin x and x are both differentiable at x=0, we can use L'Hospital's Rule.

          .

Q2. Evaluate 
Ans.
                              

Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since both are differentiable at x=0, we can use L'Hospital's Rule.

    =  

    =  

                          
    = 1.

Q3. Evaluate  .
Ans.
                                

Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since both are differentiable at x=0, we can use L'Hospital's Rule.
       =                      
     
Since after applying L'Hospital's Rule we are getting 0/0 form. So we apply the rule again till we didn't get 0/0 form.

       =                       

       =                       

       =                      

       =    
       =  120×0 
       =     0