Why is $\frac{d^2y}{dx^2} > 0$ for a minimum point and $\frac{d^2y}{dx^2} <0$ for a maximum? Also why does the second derivative not provide a reliable nature of the point of inflection?
Sorry I searched around and couldn't find any results.
[Math] Second derivative criteria for maxima and minima.
calculusderivativesmaxima-minimareal-analysis
Related Question
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Best Answer
Intuitively: if you go down and then up, then your gradient must be increasing (it's negative going down, then positive going up again). Since $\dfrac{d^2y}{dx^2}$ is the rate of change of the gradient, this means that it must be positive at a minimum value.
As for points of inflection, if you have one then your second derivative is zero. But it's possible that your second derivative can be zero without having a point of inflection, like what happens with the graph of $y=x^4$ at $x=0$. Why? Because the gradient might itself have a point of inflection!