At x=3, is the polynomial y= (4/3)x^3 -6x^2 + 11 at a maxima or minima?
First, take the first differential: y' = 4x^2 -12x. At x=3 y'= 0 so therefore the function is at a point of inflection. Taking the second derivative: y'' = 8x -12. At x=3 y''= 12. As 12 is greater than 0, the polynomial is at a minimum. If the second differential was less than 0, it would be a point of maximum and if it equaled 0 then the test fails. We must find out by comparing the sign of values of the first derivative slightly less and slightly more than the value.
JT
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