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.