# the graph of f(x)=2x^3+15x^2-84x+16 has two horizontal tangetns.One occurs at a negative value of x and the other at a positive of x. what is the negative value of x where a horizontal tangent occurs? what is the positive value of x where a horizontal tangent occurs?

The graph will have a horizontal asymptote at a local maximum, a local minimum, or possibly an inflection point. All of these can only occur at the critical points of the function -- where the first derivative is zero or fails to exist. Since a cubic polynomial is infinitely differentiable, we need only find the zeros of the first derivative:

`f(x)=2x^3+15x^2-84x+16`

`f'(x)=6x^2+30x-84`

Set `f'(x)=0` :

`6x^2+30x-84=0`

`6(x^2+5x-14)=0`

`6(x+7)(x-2)=0`

`f'(x)=0 ==> x=-7,2`

The graph of the tangent line is horizontal at x=-7 (a local maximum).

The graph of the tangent line is horizontal at x=2 (a local minimum.)

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