= 1 (x>1)
At x = 1, (1, 1) is neither a (strict) extremum nor an inflection point, even though the value of the derivatives is zero
First of all, it is necessary to specify the sufficient condition for the extremum of a divisible function
f''(x0) = 0 and f''( x0) is not equal to 0
Sufficient condition for inflection point of a differentiable function
f''(x0)=0 and f'''(x0) is not equal to 0
For your question, it should be considered like this
For a differentiable function, if x0 is a stationary point but not an extreme value point, you can consider such a situation
f''(x0)= 0 and f''(x0)=0,but we don't know whether f'''(x0) is equal to 0, so it doesn't follow necessarily from your conclusion
Your guess is obviously wrong. But the example given on the first floor isn't good either, at least in the sense that (1,1) is indeed an extreme point, which isn't enough to disprove the proposition.
The following is satisfied for the segmented function f(x)=x^4*sin(1/x), where x is not equal to 0
=0 x=0
f'(0)=0
It would be more complicated to say it in theory, so I'll say it directly in terms of the image
His image oscillates up and down the x-axis infinitely many times in an arbitrary neighborhood of x=0, sort of like the It's kind of like a sine function, except that its amplitude gets smaller and smaller, and it approaches zero infinitely, and since it's an odd function, you can think of his image in the same way as a sine function, which is obviously not an extremum, and an inflection point is defined as a concave or convex point, and any neighborhood of x=0 can change its concavity or convexity an infinite number of times, so x=0 isn't a concavity or a convexity, so x=0 is not a concavity or convexity, so x=0 isn't a convexity or an inflection point.