Are the inflection points where f'(x) = zero or where the graph changes from concave up to concave down?

Question

William Abdalla · Accepted Answer

The inflection point is a point where the graph of the function changes from concave up to concave down or vice versa. To calculate these points you have to find places where #f''(x)=0# and check if the second derivative changes sign at this point. For example to find the points of inflection for #f(x)=x^7#you have to calculate #f''(x)# first. #f'(x)=7x^6#
#f''(x)=42x^5# Now we have to check where #f''(x)=0#
#42x^5=0 iff x=0#. We found that #x=0# may be  a point of inflection. 
To find if it is such point we have to check if #f''(x)# changes sign at 0. To find this we can graph the function: graph{42x^5 [-3.894, 3.897, -1.95, 1.948]} We can see that the #f''(x)# changes sign at zero, so zero is the point of inflection.  Note
It is important to check to see whether concavity actually changes. #g(x)=x^4+3x-8# #g'(x)+ 4x^3+3# #g''(x) =12x^2# Now we have to check where #g''(x)=0#
#12x^2=0 iff x=0#. We found that #x=0# may be  a point of inflection. 
To find if it is such point we have to check if #g''(x)# changes sign at #0#. But #g''(x) =12x^2# is never negative, it is always positive or #0#, therefors the sign of #g''(x)# does not change, so there are no inflection points.

Elijah Abram · Answer

I have been taught and, following our textbook's lead, I continue to teach , that an inflection point is a point on the graph at which the concavity changes.

Using this terminology: For #f(x) = x^7#, the inflection point is #(0,0)# The function: #h(x) = 1/x# is concave down on #(-oo,0)# and concave up on #(0,oo)#. The concavity is not the same on the entire graph, but there is no inflection point, because there is no point on the graph at #x=0#.

Ezekiel Alexander · Answer

undefined The inflection points of a function occur where the second derivative changes sign, indicating a change in concavity. They are not necessarily where the first derivative is zero, but rather where the graph transitions from concave up to concave down, or vice versa.

Ezekiel Alexander · Answer

undefined Inflection points occur where the graph changes from concave up to concave down or vice versa. Therefore, they are not necessarily where the derivative, f'(x), equals zero.