Question
Sun May 01, 2011 By: Gaurav Avhad
 

why does when second derivative is less than zero the function is having maximum value can u explain me by a graph of a particular function?

Expert Reply
Mon May 02, 2011

 

Concavity and the Second Derivative Test  

 

 

 

 

 

If the second derivative of a function is greater than zero, for all x in an interval, the original graph is concave up.  

 

 

 

 

If the value of the critical number of the first derivative equals zero and the second derivative is less than zero, then the original function has a local maximum at that point.  

 

 

 

f (1) = -3

 

 

 

 

 

 

Function

 

Second Derivative

 

Conclusion

 

(-?, 1)

 

Negative

 

Concave Down

 

(1, ?)

 

Positive

 

Concave Up

 

 

Positive values mean the graph is concave up on an interval.

Negative values mean the graph is concave down on an interval.

Choose a value within each interval and recognize whether the answer is negative of positive.

To determine concavity, create a chart of intervals including the point of inflection x-value.

Therefore, a point of inflection occurs at the point (1, -3).

Substitute the x-value back into the original function to determine the order pair of the point of inflection.

f (1) =

2 ) 1() 1(3) 1(2 3

In order to determine the point of inflection set the second derivative equal to zero to isolate the x-value.

0= 6(x-1)

x=1

Find the second derivative of the function.

f? (x) = 6x-6

= 6(x-1)

Find the first derivative of the function.

f? (x) =

1 6 3 2 x x

If the value of the critical number of the first derivative equals zero and the second derivative is greater than zero, then the original function has a local minimum at that point.

Example: Find the points of inflection and determine the functions concavity.

f(x) =

2 3 2 3 x x x

 

The Second Derivative Test states:

Points of Inflection occur where the second derivative of a function equals zero or does not exist, when x is in the domain.

Points of Inflection show where a graph changes concavity.

If the derivative second of a function is less than zero, for all x in an interval, the original graph is concave down.

 

Concave Down (CD) - on an interval, the graph of a function lies below all of the tangent lines.

Concave Up (CU) - on an interval, the graph of the function lies above all of the tangent lines.

The Second Derivative Test tells you whether a function is concave up or concave down.

The Second Derivative Test evaluates concavity.

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