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I'll be writing a concise guide on inflection points soon, but I'll give you a brief overview.
Definitions (these are the two useful definitions for Specialist Maths):
1. A differentiable function \(f:D\to\mathbb{R}\) has a point of inflection at \((a,\,f(a))\), where \(a\in D\), if and only if the first derivative, \(f'\), has an isolated extremum at \(x=a\).
2. A \(k\) times (\(k\geq2\)) differentiable function \(f:D\to\mathbb{R}\) has a point of inflection at \((a,\,f(a))\), where \(a\in D\), if and only if the concavity (curvature) of \(f\) changes sign at \(x=a\).
In regards to the above question, could someone explain how arctan has a point of inflection, even though we cannot solve its derivative for zero? What am I missing?
A point of inflection does
not simultaneously also have to be a stationary point. (If it is, we call it a "stationary point of inflection", but otherwise, it is a "non-stationary point of inflection).
And in the same way, how come the point of greatest gradient is at x=0 if the gradient is zero at this point, but is positive elsewhere? My main point of confusion is that if the derivative of arctan is 1/1+x^2, where the derivative is always greater than zero, how come we have a point of inflection?
Same as above. Points of inflection do not have to have a 0 gradient. The fact that \[\dfrac{d}{dx}\Big[\arctan(x)\Big]=\dfrac1{1+x^2}>0\ \ \ \forall x\in\mathbb{R}\] bears no affect on whether the graph of \(y=\arctan(x)\) has a point of inflection or not.
The graph of \(g(x)=\arctan(x)\) has a point of inflection at \((0,\,0)\) because the graph of the first derivative, \(g'\), has a local maximum at \(x=0\). Or, using the second definition, the graph of \(g\) changes concavity (from concave up to concave down) at \(x=0\).
My reasoning for why the point of greatest gradient is at x=0 is that the reciprocal of x^2 +1 has a maximum value at x=0 and hence maximum gradient is at this point.
You are correct. The fact that \(g'\) is maximum at \(x=0\) makes this the point of steepest ascent as well as a point of inflection.