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Yes, a point of inflection exists where f''(x) = DNE.
We know that the point of inflection, also known as the inflection point, is where the function's concavity changes. Changing the function from concave down to concave up, or vice versa, means that. In other words, an inflection point is a location at which the rate of slope change from an increasing to a decreasing manner, or vice versa, occurs. There is no doubt that those points are neither local maxima nor minima. They are called stationary points.
Let's take an example [tex]f(x) = x^{\frac{1}{3} }[/tex]
Here, [tex]f'(x)=\frac{1}{3x^{\frac{2}{3} } }[/tex]
and [tex]f''(x)=-\frac{2}{9x^{\frac{5}{3} } }[/tex]
Clearly f''(x) = DNE at x = 0. We are going to check if x = 0 is a point of inflection or not.
At x = -1, (left side of 0), [tex]f''(-1)=-\frac{2}{9(-1)^{\frac{5}{3} } }=\frac{2}{9} > 0[/tex] (concave up)
At x = 1, (right side of 0), [tex]f''(1)=-\frac{2}{9(1)^{\frac{5}{3} } }=-\frac{2}{9} < 0[/tex] (concave down)
Then by the definition, we can conclude that x = 0 is a point of inflection.
Therefore, a point of inflection exists where f''(x) = DNE.
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