No, because f '(x) has to be continuous at x for it to be counted as an inflection point.
While it may be true that f '(x) to the right and to the left are of different signs, since at a sharp point, f '(x) does not exist, it is not considered as an inflection point.
While it may be true that f '(x) to the right and to the left are of different signs, since at a sharp point, f '(x) does not exist, it is not considered as an inflection point.
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There are different definitions of inflection point in different books, and they are not all equivalent. Even Wikipedia has several definitions which they say are equivalent, but they are not. So, you have to answer this question based on the definition in your calculus book, or the definition you were given in class.
My opinion is that an inflection point is a point on the curve y=f(x) at which the first derivative is defined and the such that the concavity is opposite on the right of that point to the concavity on left. In other words as you increase x from the left side of the point to the right side of the point the concavity changes from concave up to concave down or vice versa. Notice that f''(x) is not required to exist at such a point, but that in order to determine concavity, f''(x) must exist on both sides of the point.
An example would be f(x) = x^2/2 for x>=0 and -x^2/2 for x<0. This function has an inflection point at x=0 even though f''(x) is not defined there. In fact f'(x) = |x| so the first derivative has a sharp point at x=0 just as in your question. Nonetheless, the concavity of y=f(x) changes at x=0 so there is an inflection point there.
But, in your class inflection point might have been defined differently, and then you have to go with that.
My opinion is that an inflection point is a point on the curve y=f(x) at which the first derivative is defined and the such that the concavity is opposite on the right of that point to the concavity on left. In other words as you increase x from the left side of the point to the right side of the point the concavity changes from concave up to concave down or vice versa. Notice that f''(x) is not required to exist at such a point, but that in order to determine concavity, f''(x) must exist on both sides of the point.
An example would be f(x) = x^2/2 for x>=0 and -x^2/2 for x<0. This function has an inflection point at x=0 even though f''(x) is not defined there. In fact f'(x) = |x| so the first derivative has a sharp point at x=0 just as in your question. Nonetheless, the concavity of y=f(x) changes at x=0 so there is an inflection point there.
But, in your class inflection point might have been defined differently, and then you have to go with that.