How to find the minimum or maximum of a cubic function?

Hi,

y = x^3 - 9x^2 + 19x - 3

differentiate wrt x we get,
y ' = 3x^2 - 18x + 19

again differentiate wrt x we get,
y'' = 6x - 18

For Minimum or Maximum value of y : y' = 0

From y' = 0
=> 3x^2 - 18x + 19 = 0
=> x = {18 +/- sqrt[(- 18)^2 - 4.3.19]}/2.3
=> x = {18 +/- sqrt[324-228]}/6
=> x = {18 +/- sqrt(96)}/6
=> x = (18 +/- 9.8)/6

Taking (+ve) sign,
x = (18 + 9.8)/6 = 4.6

Taking (-ve) sign,
x = (18 - 9.8)/6 = 8.2/6 = 1.4

Now, [y''] at x=4.6 = 6*4.6 - 18 = 27.6 - 18 = 9.6 > 0
and [y''] at x=1.4 = 6*1.4 - 18 = 8.4 - 18 = - 9.6 < 0

So, y has a minimum value at x = 4.6
and y has a maximum value at x = 1.4

Minimum value = (4.6)^3 - 9*(4.6)^2 + 19*(4.6) - 3
= 97.336 - 190.44 + 87.4 - 3
= - 8.704

Maximum value = (1.4)^3 - 9*(1.4)^2 + 19*(1.4) - 3
= 2.744 - 17.64 + 26.6 - 3
= 29.344 - 20.64
= 8.704

Hope this helps u:)