Is a critical value always a critical point as well?

Question

My impression from reading the definition of critical points is that the function has to be defined there, and its derivative at that value has to be either zero or undefined.

However, in a problem assigned me by my teacher, which is to find the critical points of the function $x(x-4)^3$, his answer says that the function has 2 critical values $x = 1, x = 4$, but only one critical point $(1, -27)$

I think something is wrong here because based on my understanding, if something is a critical value, it is also by definition a critical point.

Answer 1

There is a very small difference between the two.

Critical points are defined only in the domain of the derivative i.e. only when the function is differentiable. So in the case of your function, we find only one critical point, at $x=1$.

But critical values are all those values, where a maxima, minima, or a point of inflection can be found, not necessarily having a zero derivative.

By definition, all critical points are critical values but not all critical values are critical points.

Answer 2

The answer is partly hinted at already in your question. The critical point is a point in the plane where you would plot the graph of the function. In general you would write that as the point $(x, f(x))$ or in your case, when $x=1$ and $f(x) = -27$ that is the point $(1,-27)$. The critical value should be the just the $x$ component. So in general when $(x, y)$ is a critical point of $f$ then $x$ is a critical value of $f$.

This function has a different thing at play here. Look at the plot! At the critical value 4 there is neither a minimum nor a maximum, while at the critical value 1, there is a minimum. I think this is the differentiation your teacher wanted to make. There are two values, and critical points, but only one local minimum/maximum.