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Lecture 16

     

 MATH 121 (Wg Cdr Monir) 

Maxima and Minima


    Some Definitions:
    ♦ Increasing Function: 
If  y=f(x) is a function of x and y increases as x increases in a certain interval, Then y is called an increasing function of x in that interval.

  dydx=tanΨ>0Ψ<90  

    ♦ Decreasing Function:
If y=f(x) is a function of x and y decreases as x increases in a certain interval, then y is called a decreasing function of x in that interval.

  dydx=tanΨ<0Ψ>90  

    ♦ Concavity:  
If the graph of a function f lies above all of its tangents on an interval l, then it is said to be concave up on l.
If the graph of a function f lies below all of its tangent on an interval l, then it is said to be concave down.


    ♦ Test for concavity
• The function f, is concave up on any open interval l where f(x)>0 and it is concave down where f(x)<0 
• Inflection Point: If x=a inflection point, when
        (1)  f(x)>0 when x<a and f(x)<0 when x>a 
    or
        (2) f(x)<0 when x<a and f(x)>0 when x>a.

    Note: Inflection point does not occur where f(x)=0. The inflection point occurs where the sign of the second derivative change.


Question: Find the maximum and minimum value of the function  x55x4+5x31. Also, find the inflection point if any.
Solution: Let,  f(x)=x55x4+5x31f(x)=5x420x3+15x2  

For maximum or minimum f(x)=0
  5x420x3+15x2=0or, 5x2(x24x+3)=0or, 5x2(x1)(x3)=0or, x=0,1,3  
Now,  f(x)=20x360x2+30x 
For x = 1
  f(x)=2060+30=10<0
then the function f(x) is maximum at x = 1 and the maximum value is,                        f(x)=155×14+5×131=0                        

For x = 3
  f(x)=20×3360×33+30×3=90>0  
then the function f(x) is minimum at x = 3 and the minimum value is,  f(x)=x55×34+5×331=28  

For x = 0
  f(x)=20×060×0+30×0=0f(x)=20x360xpe2+30xf(x)=60x3120x+30Now for   x=0,f(x)=300  
the function has neither maximum nor minimum at x = 0
At x = 0 is an inflection point



         

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