How do you find points of inflection and determine the intervals of concavity given #y=3/(x^2+4)#?

Answer 1

Below

#y=3/(x^2+4)# #dy/(dx)=-(6x)/(x^2+4)^2# #(d^2y)/(dx^2)=-6times(4-3x^2)/(x^2+4)^3#
For stationary points, #dy/(dx)=-(6x)/(x^2+4)^2=0#
ie #-6x=0# #x=0#
Test #x=0# #(d^2y)/(dx^2)=-3/8 <0#
Therefore, it is a maximum and concave down at #x=0# #(0,3/4)#
For point of inflexion, #(d^2y)/(dx^2)=0#
#-6times(4-3x^2)/(x^2+4)^3=0#
#4-3x^2=0# #4/3=x^2# #x=+-2/sqrt3#
Test when #x=-2/sqrt3#
#x=-1.5# #(d^2y)/(dx^2)=1056/15625#
#x=-2/sqrt3# #(d^2y)/(dx^2)=0#
#x=-1# #(d^2y)/(dx^2)=-6/125#
Therefore, there is a change in concavity so there is a point of inflexion at #x=-2/sqrt3# #(-2/sqrt3,9/16)#
Test when #x=2/sqrt3#
#x=1# #(d^2y)/(dx^2)=-6/125#
#x=2/sqrt3# #(d^2y)/(dx^2)=0#
#x=1.5# #(d^2y)/(dx^2)=1056/15625#
Therefore there is a change in concavity so there is a point of inflexion at #x=2/sqrt3# #(2/sqrt3,9/16)#
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Answer 2

To find points of inflection and determine intervals of concavity for ( y = \frac{3}{x^2 + 4} ), follow these steps:

  1. Find the second derivative of the function.
  2. Set the second derivative equal to zero and solve for ( x ).
  3. Determine the sign of the second derivative in the intervals defined by the critical points found in step 2.
  4. Points of inflection occur where the sign of the second derivative changes.
  5. Use the sign of the second derivative to determine the intervals of concavity.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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