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The maximum value of f(x) = x3 – 3x2 – 9x + 2 on the interval [0, 6] is
2
56
135
There is no maximum on [0, 6]
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Take the derivative of f(x). It is
f'(x) = .
Equal it to zero and find the roots:
= ,
= ,
(x+1)*(x-3) = 0.
The roots are x = -1, x = 3.
They are the candidates for maximum/minimum of f(x).
To check further, look into the second derivative of f(x) at these values of x.
f''(x) = 6x - 6;
f''(-1) = 6*(-1) -6 = -12; f''(3) = 6*3-6 = 12.
Hence, x = -1 is the maximum, and x = 3 is the minimum.
The plot of f(x) below confirms this analysis.