Can a critical point be neither a max or min?

One example is the simplest cubic function, y=x^3 , which has a single critical point at x=0. (Both the first and second derivatives equal 0 there, so it is also an inflection point.) The function has neither a minimum nor a maximum anywhere, so the (sole) critical value is neither a minimum nor a maximum.

Is a critical point always a maximum or minimum?

If c is a critical point for f(x), such that f ‘(x) changes its sign as x crosses from the left to the right of c, then c is a local extremum. is a local maximum. So the critical point 0 is a local minimum. So the critical point -1 is a local minimum.

Can there be no max or min?

It is completely possible for a function to not have a relative maximum and/or a relative minimum. Here is the graph for this function. In this case we still have a relative and absolute minimum of zero at x=0 x = 0 .

Can a critical value be undefined?

Critical points of a function are where the derivative is 0 or undefined. Remember that critical points must be in the domain of the function. So if x is undefined in f(x), it cannot be a critical point, but if x is defined in f(x) but undefined in f'(x), it is a critical point.

Can critical points be negative?

If the derivative is negative to the left of the critical point and positive to the right of it, the graph has a local minimum at that point (and it’s possible this local minimum might be a global minimum).

Is a global maximum always a critical point?

Fact: Critical points are candidate points for both global and local extrema. If f is continuous on a closed, bounded set S, then f attains both a global max value and a global min value there.

How do you know if a critical point is maximum or minimum or saddle point?

If D>0 and fxx(a,b)<0 f x x ( a , b ) < 0 then there is a relative maximum at (a,b) . If D<0 then the point (a,b) is a saddle point. If D=0 then the point (a,b) may be a relative minimum, relative maximum or a saddle point. Other techniques would need to be used to classify the critical point.

Is a relative maximum a critical point?

A function f(x) has a relative maximum at x = a if there is an open interval containing a such that f(a) ≥ f(x) for all x in the interval. If f(x) has a relative minimum or maximum at x = a, then f (a) must equal zero or f (a) must be undefined. That is, x = a must be a critical point of f(x).

Can a function have no minimum value?

A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum. If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.

Can there be no critical points?

A2A, thanks. If it has no critical points, it is either everywhere increasing or everywhere decreasing. (Otherwise, if it was decreasing on one part of the domain, and increasing on another part, then the boundary point between these two parts would be a critical point.)

Can local max and min be undefined?

Fermat’s Theorem says that the only points at which a function can have a local maximum or minimum are points at which the derivative is zero or the derivative is undefined.