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