SOLUTION: Please need assistance 1) Consider the function f (x) = x square - 4x cube (a) Find the stationary points of . (b) Use the First Derivative Test to determine any local ma

Question 1116348: Please need assistance

1) Consider the function f (x) = x square - 4x cube
(a) Find the stationary points of .
(b) Use the First Derivative Test to determine any local maximum or local minimum of .

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
f(x) = x^2 - 4x^3
:
the first derivative f'(x)
:
f'(x) = 2x -12x^2
:
to find stationary points set f'(x) = 0
:
2x -12x^2 = 0
:
12x^2 -2x = 0
:
x^2 -x/6 = 0
:
complete the square
:
x^2 -x/6 +1/144 = 1/144
:
(x -1/12)^2 = 1/144
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x -1/12 = sqrt(1/144)
:
x = 1/12 +1/12 = 1/6
x = 1/12 -1/12 = 0
:
f(1/6) = (1/6)^2 - 4(1/6)^3
:
f(1/6) = (1/36) -4(1/216) = 6/216 -(4/216) = 2/216 = 1/108
:
f(0) = 0^2 -4(0)^3 = 0
:
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1.a stationary points are (0,0) and (1/6, 1/108)
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1.b the second derivative is used to determine local maxima and local minima
:
f''(x) = 2 -24x
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The sign of the second derivative at the stationary point is positive for a local minimum, and negative for a local maximum.
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f''(0) = 2 and f''(1/6) = 2 -4 = -2, therefore
:
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(0,0) is a local minimum and (1/6, 1/108) is a local maximum
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:

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