# 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.

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.

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

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