# why does a parabola have 3 critical points when finding minimum distance? What do they represent?

A function is given 6-x^2. The point is (0,3). After setting d'(x) to zero I get three critical points 0, -sqrt(2.5) and sqrt(2.5)

### 4 Answers

- PopeLv 71 month agoFavorite Answer
Did you try graphing it? Even a very rough graph would explain the result.

Here the parabola is the curve y = 6 - x². From point (0, 3) there are three normals, meeting the curve at these three points:

(-√(10)/2, 7/2)

(0, 6)

(√(10)/2, 7/2)

At the first and third points the distance is minimized. The second point is the vertex, representing a local maximum for the distance.

- davidLv 71 month ago
WHY? All given points will NOT have 3 critical points. This given point is on the axis f symmetry of the parabola ... that is why it has 3 crit. points ... 2 are symmetric (because of axis of summetry) the 3rd might be a max or a min dist or neither.

Pretty much. A point on the axis does not necessarily have three normals. It sometimes has only one.There is always at least one local min, sometimes two. There is never more than one local max, sometime zero. There are never more than three normals. One normal may be neither max nor min.

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- PhilipLv 61 month ago
f(x) = 6-x^2, f'(x) = -2x, f'(x) = 0 ---> x = 0, tangent line L, say, at P(0,6) has slope = 0.

The point Q(0,3) is on a line M, say, which is normal to f(x) at P. PQ = 3 and is the

minimum distance from Q to f(x).

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- ted sLv 71 month ago
my question to you is HOW does a linear function have 3 roots ?....and (0,3) is not on the graph of the function !!! ...TRY AGAIN....ahhh..you have a distance function from ( 0 , 3) to the curve...ignore my comments

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what’s your point capitain other just unnecessary argument?

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Great answer