# 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 . What is the maximum height of the bird.

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The distance of the bird from its nest is the magnitude of a vector which is a non-negative number.

The height of the bird (y) is a function of its distance from its nest (x). The relation between the two is y = 3x^3 + 2x^2 - 10x + 5.

The height of the bird has a maximum value at the solution of y' = 0, x = a and if y''(a) is negative.

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

y' = 0

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

=> `x = (-4 +- sqrt(16 + 360))/18`

=> `x = (-4 +- sqrt 376)/18`

=> `x = (-2 +- sqrt 94)/9`

y'' = 18x - 4

At `x = (-2 + sqrt 94)/9` , y'' is positive and at `x = (-2 - sqrt 94)/9` , y'' is negative. But the distance of the bird from its nest cannot be negative. As the second derivative of y is not negative for a valid value of x, there is no maximum limit to the height of the bird.

**There is no maximum value to the height at which the bird flies.**

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

`y=3x^3+2x^2-10x+5`

`=(x-1)(3x^2+5x-5)`

when y=15 (aprox) ,when possible distance of the bird is x=-1.5 (approx) here it is distance not displacement .

But distance can not be negative so y=15 not possible to consider as local maxima . Also in this particular problem global maxima does not exist as x increases beyond y also increases .

**Thus we conclude that maximum height height of the bird is an infinite (very large in comparison ) . Thus we can conclude maximum height bird will never attained or does not exist.**

Why is the distance of the bird from the nest only positive.