How do you find all points of inflection given #y=-x^4+3x^2-4#?

Question

Emilia Aguilar · Accepted Answer

#(1,-2) and (-1,-2)" " #are two inflection points of the given function.

The inflection points of a function is determined by computing #" "# the second derivative of #y# then solving for #color(blue)(y''=0)#. #" "# #y=-x^4+3x^2-4# #" "# #y'=-4x^3+6x# #" "# #y''=-12x^2+6# #" "# To find the inflection points we would solve the equation: #" "# #color(blue)(y''=0)# #" "# #rArr-12x^2+6=0# #" "# #rArr-12(x^2-1)=0# #" "# #rArr-12(x-1)(x+1)=0# #" "# #rArrx-1=0rArrx=1" "# #" "# Or #" "# #x+1=0rArrx=-1# #" "# The ordinate of the point of abscissa #x=1# is: #" "# # y_((x=1))=-1^4+3(1)^2-4=-1+3-4=-2# #" "# The ordinate of the point of abscissa #x=-1# is: #" "# # y_((x=-1))=-(-1)^4+3(-1)^2-4=-1+3-4=-2# #" "# Hence, #" "# #(1,-2) and (-1,-2)" "# are two inflection points of the given function.

Isabella Ackerman · Answer

undefined To find the points of inflection of the function y = -x^4 + 3x^2 - 4, follow these steps:

1. Take the second derivative of the function.
2. Set the second derivative equal to zero and solve for x to find the potential points of inflection.
3. Determine the concavity of the function in the intervals defined by the potential points of inflection.
4. Confirm the concavity by examining the sign of the second derivative around each potential point of inflection.
5. Identify the points of inflection where the concavity changes.

Let's go through these steps:

1. Find the second derivative of the function y = -x^4 + 3x^2 - 4:
   y' = -4x^3 + 6x
   y'' = -12x^2 + 6

2. Set the second derivative equal to zero and solve for x:
   -12x^2 + 6 = 0
   x^2 = 1/2
   x = ±√(1/2)
   So, the potential points of inflection are x = √(1/2) and x = -√(1/2).

3. Determine the concavity of the function in the intervals defined by the potential points of inflection:
   Use test points within each interval to determine the sign of the second derivative.

4. Confirm the concavity by examining the sign of the second derivative around each potential point of inflection:
   Substitute test points into the second derivative:
   For x < -√(1/2): Choose x = -1
   y''(-1) = -12(-1)^2 + 6 = -6
   For -√(1/2) < x < √(1/2): Choose x = 0
   y''(0) = -12(0)^2 + 6 = 6
   For x > √(1/2): Choose x = 1
   y''(1) = -12(1)^2 + 6 = -6

5. Identify the points of inflection where the concavity changes:
   At x = √(1/2), the concavity changes from negative to positive, indicating a point of inflection.
   At x = -√(1/2), the concavity changes from positive to negative, indicating another point of inflection.

So, the points of inflection for the function y = -x^4 + 3x^2 - 4 are ( √(1/2), f( √(1/2))) and (-√(1/2), f(-√(1/2))), where f(x) = -x^4 + 3x^2 - 4.