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Question
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x3 − 6x2 + 9x + 15
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Solution
Given function, f(x) = x3 - 6x2 + 9x + 15
`therefore` f'(x) = 3x2 - 12x + 9
= 3 (x2 - 4x + 3)
= 3 (x - 1)(x - 3)
=> x = 1 or x = 3
`therefore` x can have lowest or highest value at x = 1 or x = 3.
f''(x) = 3(2x - 4) = 6x - 12
At, x = 1, f''(x) = 6 × 1 - 12 = - 6 (Negative)
∴ The value of the function at x = 1 is a local maximum.
Maximum value = f(1) = (1)3 - 6(1)2 + 9(1) + 15
= 1 - 6 + 9 + 15
= 19
At x = 3, f'' = 6 × 3 - 12 = 6 positive
∴ f(x) has a local minimum at x = 3.
Minimum value = f(3) = (3)3 - 6(3)2 + 9(3) + 15
= 27 - 54 + 27 + 15
= 15
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