Wednesday 20 February 2019

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.


Concave and convex

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.


Convex


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.


Concave curve


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.

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