>The question is the same as the following: > >Let f(x)=x^2for x<=0 0 for 0<=x<=1 and (x-1)^2 for x>1 > >Does the function have a relative minimum?
The function has a relative minimum at each point of the interval [0, 1] but the first and second derivative tests fail.
A relative minimum of a function f occurs at a point c iff f(c) <= f(x) for all x in some open interval containing c.
However no critical point is situated such that f'(x) to the left is negative while f'(x) to the right is positive nor is there any critical point c so that f''(c) is positive. Since the definition of a relative minimum is satisfied, f(c) IS a relative minimum. Since the conditions of the theorems for the first and second derivative tests are not met, they are inconclusive.
Contrasted with the original question on inflection points, the definition of an inflection point is not met, i.e.,
A point (c, f(c)) is called a point of inflection for the graph y = f(x) iff there is an open neighborhood containing c over which f''(x) is positive on one side of c and negative on the other side.
Since f''(x) = 0 on the segment connecting the two curved portions of the graph, the definition doesn't hold and the curve has no inflection points.