A bird is flying at height y that is a function of its distance from a tree x given by y = 3x^3 + 2x^2 - 10x + 5 . Is there a minimum height of the bird.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The height at which the bird flies (y) is related to the distance from its nest by y = 3x^3 + 2x^2 - 10x + 5.

The graph of the relation is:

The distance of the bird from its nest is a positive number (as this is the magnitude of a vector). From the graph it is evident that there is no limit to how large y can be for positive values of x.

To determine the lowest value of y, calculus can be used to determine y' and y''. Solve y' = 0 for x

y' = 9x^2 + 4x - 10

y' = 0

=> 9x^2 + 4x - 10 = 0

The positive root of the equation is `(sqrt 94 - 2)/9` . At this point the value of y is negative and equal to -0.2128.

But the height of a bird (using logic) cannot be negative, it cannot fly below the surface.

In case the height can take on negative values, the minimum height of the bird is -0.2128

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llltkl | College Teacher | (Level 3) Valedictorian

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The height y, refers to height of the bird from the tree (hopefully the tree-top). Therefore, it can take on negative values (of course in the range of the actual height of the tree). The bird can definitely fly at a level, a bit lower than the tree-top.

The minimum height of the bird (with respect to the tree-top) is thus -0.2128 m.

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aruv | High School Teacher | (Level 2) Valedictorian

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With help of calculus we can  define minimum and maximum local max or local min . Some time global max /min. Here global max /min does not exist . Local min /max exist.But it is physical problem so non negative restriction will effect the results.It is simple to observe from graph and then logical conclusion.  Let first draw the graph

when y=0 ,possible distance of the bird is x=1 , x=.61 and x=-2.28

distance can not be negative  so neglect the value of x=-2.28

Thus when x=1 or .61  ,y=0

Thus we conclude  that minimum height of the bird  is zero.