What are the points of inflection of #f(x)=x/lnx

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Question

What are the points of inflection of #f(x)=x/lnx

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Charles Arocha · Accepted Answer

#x in{1,e^2}#

Find where #f''(x)=0# OR #f''(x)# switches from negative to positive instantaneously (like with an asymptote).

First, we find points where #f''(x)=0#.

#f'(x)=(lnx-x(1/x))/(lnx)^2=1/(lnx)-1/(lnx)^2#

#f''(x) = -1/(x(lnx)^2) +2/(x(lnx)^3)#

Set this to 0 now.

#0 = -1/(x(lnx)^2) +2/(x(lnx)^3)#

#0 = -lnx+2#

#lnx = 2#

#x = e^2#

This shows that the only solution to #f''(x)=0# is #x=e^2#. Additionally, in order to be a point of inflection, the graph of #f''(x)# must cross the x-axis from positive to negative or negative to positive AT #x=e^2# (which can be proven by showing that #d/dx(x/lnx)!=0# when #x=e^2#, but I will not include this proof unless it is requested).

So, #x=e^2# is a point of inflection.

We also need to look for points of instantaneous change. In this case, we have one: the asymptote at #x=1#.

At #x=1#, #ln(x)=0#, and it switches from negative to positive by definition. This means that #-1/(x(lnx)^2) +2/(x(lnx)^3)# will also jump from negative to positive, since the denominator is switching signs. Even though #f''(x)# never equals 0, it switches signs at #x=1#, so #x=1# is a point of inflection.

Therefore the two points of inflection for #f(x)=x/lnx# are #x=1# and #x=e^2#

Charles Arocha · Answer

undefined To find the points of inflection for the function $ f(x) = \frac{x}{\ln x} $, follow these steps:

1. Find the second derivative of $ f(x) $.
$$ f''(x) = \frac{d^2}{dx^2}\left(\frac{x}{\ln x}\right) $$

2. Determine the critical points of $ f''(x) $ by setting $ f''(x) $ equal to zero and solving for $ x $.

3. Test the concavity of the function around the critical points by evaluating $ f''(x) $ in the intervals determined by the critical points.

4. The points where the concavity changes (i.e., where $ f''(x) = 0 $ or does not exist) are the points of inflection.

5. Finally, check that the second derivative changes sign across these points to confirm they are points of inflection.