# Calculus Critical Point Confusion - 2nd Derivative Confusion

I am working with the function:

$$f(x) = 4x^5 - 3x^4 + 2x^3 - x^2 + 3$$.

One of the critical points $$f'(x) = 0$$ occurs when $$x=0$$. Checking the second derivative, we see that $$f''(0) = -2$$. This would lead one to the conclusion that the point $$x=0$$ is a local maximum.

However, looking at the plot of the function, we see that $$x=0$$ looks more like a saddle: So, I am just wondering how we could conclude it is a saddle point if the second derivative is (incorrectly?) saying it is a local max.

Thanks.

• Try plotting it for $-1\le x\lt 1$ – AgentS Nov 22 '19 at 0:39
• this is one of those times when looks can be deceiving :) – peek-a-boo Nov 22 '19 at 0:42

You have to zoom in more to understand the whole picture:

At first glance it might seem like a saddle point: Let’s zoom in more This is starting to look like more of max than a saddle so let’s zoom in a little bit more: This is definitely a maximum according to our last picture.

Remember that even though it is a small interval and the function increases almost immediately after it, there are infinite points around $$0$$ such that they give a lower value for our function which makes $$f(0)$$ greater than its neighbours.

Source: Desmos Graphing Calculator

The graph can lead to bad evaluation, indeed in this case fince $$f'(0)=0$$ and $$f''(0)<0$$ the point is of course a local maximum for the function. Graphs can deceive. You have $$f(0)=3$$ and the derivative is $$f'(x)=20x^4-12x^3+6x^2-2x$$ Nothing here hints to the function having an inflection point at $$0$$. To the contrary, $$f'(x)=2x(10x^3-6x^2+3x-2)$$ is positive for $$-\delta and negative for $$0 (some $$\delta>0$$), because the term in parenthesis is $$-2$$ for $$x=0$$.

Therefore $$0$$ is a point of local maximum, which is confirmed by $$f''(x)=80x^3-36x^2+12x-2$$ and $$f''(0)=-2$$.

In order that $$c$$ is an inflection point for a polynomial function $$f$$, one needs that $$c$$ is a root of multiplicity greater than one and odd of the polynomial $$f(x)-f(c)$$: just write the Taylor expansion at $$c$$ to realize it.