find the extreme values of the following y = X3 - 6X2 + 9X - 8
Let's first find the derivative of the function:
Function have the extreme values when "\\dfrac{dy}{dx}=0":
This quadratic equation has two roots: "x_1=3," "x_2=1". Therefore, the extreme values of the function are: "x=1" and "x=3". Let's also find the nature of the extreme values. Let’s divide the function into the following intervals: "x<1", "1<x<3," "x>3". Then, we can choose the test points in these intervals: "x=0", "x=2" and "x=4." Finally, we can determine the sign of the derivative in these intervals:
As we can see from calculations, derivative has sign plus on interval "x<1" and sign minus on interval "1<x<3". Since derivative change its sign from positive to negative, the extreme value at point "x=1" is the maximum. Also, we can see that derivative has sign plus on interval "x>3". Since derivative change its sign from negative to positive, the extreme value at point "x=3" is the minimum.
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