For the curve $y=(2x-1)^4$, the derivative $8(2x-1)^3$ shows that the only stationary point is $(0.5, 0)$, which means it just touches the $x$-axis at that point. However, when determining the nature of the stationary point, the second derivative is $48(2x-1)^2$. Substituting $x=0.5$ into the second derivative gives $\frac{d^2y}{dx^2}=0$, which implies the stationary point is a point of inflection. However, when drawing the graph, it is actually a minimum point.
I also checked it by substituting $x=0.49$ and $x=0.51$ into the first derivative, which gives a negative and positive gradient, respectively, showing a minimum point.
So why does the second derivative show a point of inflection?
Shouldn't it be positive, which shows a minimum point?
Is it possibly to do with the fact that the curve does not actually cross the $x$-axis but only touches it, or is that irrelevant?
Best Answer
Just like $f'(x_0)=0$ doesn't mean $x_0$ is a point of maximum or minimum, $f''(x_0)$ doesn't mean $x_0$ is a point of inflection.
The usual example for the first case is $f(x)=x^3$: $f'(0)=0$, but $0$ is a point of inflection and neither a point of maximum nor a point of minimum.
The condition that $f''(x_0)=0$ when $x_0$ is a point of inflection is only necessary and not sufficient (for points where the second derivative exists).
Your function is essentially the same as $f(x)=x^4$, which obviously has a minimum at $0$.
For “well behaved” functions (more precisely, analytic, but the condition can be relaxed), a point of inflection is where the first derivative changes from increasing to decreasing or conversely: so they're the points where the first derivative has a maximum or minimum.
For $f(x)=x^4$, we have $f'(x)=4x^3$, and this function has neither a maximum nor a minimum at $0$.
When you're looking for the points of inflection, you certainly find the points where the second derivative vanishes. Among them there are the inflection points and you have to check for the sign of the second derivative at either side of the point.
A different method is to look at the lowest $n$ such that $f^{(n)}(x_0)\ne0$: if $n$ is even the point $x_0$ is either a maximum or a minimum; if $n$ is odd the point $x_0$ is a point of inflection (assuming $f''(x_0)=0$ to begin with).
However, this method might not be conclusive. Consider $$ f(x)=\begin{cases} 0 & x=0 \\ \exp(-1/x^2) & x\ne0 \end{cases} $$ Then $f^{(n)}(0)=0$ for every $n$; $0$ is obviously a point of minimum.
The function $$ F(x)=\int_{0}^x f(t)\,dt $$ has the same property: $F^{(n)}(0)=0$ for all $n$, and $F$ has a point of inflection at $0$.
On the other hand, the function $$ f(x)=\begin{cases} 0 & x=0 \\ x^4\sin(1/x) & x\ne0 \end{cases} $$ has $f''(0)=0$, but the second derivative takes positive and negative values in every left and right neighborhood of $0$, so $0$ is not a point of inflection, but it's not a point of maximum or minimum either.
There can be other points of inflection, though: the function $f(x)=\sqrt[3]{x}$ has an inflection at $0$, where it is not differentiable.