What is the smallest possible slope for a tangent to the graph of the equation

`y=x^3-30x^2+32x`

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You need to evaluate the derivative of the given function to evaluate the slope of the tangent line to the curve representing the function, such that:

`(dy)/(dx) = 3x^2 - 60x + 32`

You should find the smallest value of the quadratic equation, hence, you need to evaluate the y coordinate of the vertex of parabola representing quadratic equation `(dy)/(dx) = 3x^2 - 60x + 32` , such that:

`y = (-Delta)/(4a)`

`Delta = b^2 - 4ac`

Identifying the coefficients `a,b,c` yields:

`a = 3, b = -60, c = 32`

`Delta = 3600 - 12*32 = 3216`

`y = (-3216)/12 => y = -268`

**Hence, evaluating the smallest value of the slope of the tangent line to the given curve, yields `y = -268.` **

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