We have said that a point of inflection occurs when

Up to this point, we have only looked at stationary points satisfying this condition. Let's now look at the curve.

If we differentiate this, we get

There are no real values of x for which this equals nought, so there are no stationary points. If we differentiate again, we get

which clearly equals nought at x = 1/3. The third derivative is

so at x = 3, according to our rule, there is a point of inflection. To see what this means, we need to look at a graph of the function.

You can see that the tangent to the curve at this point crosses through the curve.

A point of inflection, is a point on a curve, where the gradient is not changing. This means that the curve is actually a "straight line" at this point. The direction that the curve is moving in actually changes at a point of inflection. In our example, the general trend of the curve, to the left of the PoI was to "bend over." After the PoI, the curve is tending to "bend up."( You  might think of the two situations as being convex and concave, respectively.)

In the case of the y = x⁴ curve, the graph continued to be concave as it passed through x = 0.

In summary

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