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.

Answered by James T. Maths tutor

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