Does the function f(x) = x^3+3*x^2-9*x have a minimum and maximum value?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to determine the maximum and minimum value of the function: f(x) = x^3+3*x^2-9*x

Here, we can only determine the local extreme points.

The extreme points of a function f(x) lie where f'(x) = 0.

The maximum point is where f''(x) is negative and the minimum point is where f''(x) is positive.

f'(x) = 3x^2 + 6x - 9 = 0

=> f'(x) = x^2 + 2x - 3 = 0

=> x^2 + 3x - x - 3 = 0

=> x(x + 3) - 1(x + 3) = 0

=> (x - 1)(x + 3) = 0

=> x = 1 and x = -3

f''(x) = 6x + 6

f''(1) = 12

Therefore we have a minimum value at x = 1

At x = 1, f(1) = 1 + 3 - 9 = -5

f''(-3) = -18 + 6 = -12

Therefore we have a maximum value at x = -3

At x = -3, f(-3) = (-3)^3 + 3*9 + 9*3 = 27

The local maximum value of the function is 27 and the local minimum value is -5.

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