Which points are the best approximations of the relative maximum and minimum of the function? f(×)=×^3+3×^2-9×-8
To find the relative maximum and minimum of the function f(x) = x^3 + 3x^2 - 9x - 8, we need to find the critical points and determine whether they correspond to maximum or minimum values.
First, let's find the derivative of f(x) with respect to x:
f'(x) = 3x^2 + 6x - 9
To find the critical points, we set the derivative equal to zero and solve for x:
3x^2 + 6x - 9 = 0
Dividing through by 3:
x^2 + 2x - 3 = 0
Factoring the quadratic equation:
(x + 3)(x - 1) = 0
Setting each factor equal to zero and solving for x gives us the critical points:
x + 3 = 0 or x - 1 = 0
x = -3 or x = 1
Now, we need to determine whether these critical points correspond to maximum or minimum values. We can do this by checking the values of the second derivative:
f''(x) = 6x + 6
Evaluating the second derivative at the critical points:
f''(-3) = 6(-3) + 6 = -18 + 6 = -12
f''(1) = 6(1) + 6 = 6 + 6 = 12
Since f''(-3) < 0, the critical point x = -3 corresponds to a relative maximum. Since f''(1) > 0, the critical point x = 1 corresponds to a relative minimum.
Therefore, the best approximations of the relative maximum and minimum of the function f(x) = x^3 + 3x^2 - 9x - 8 are:
Relative Maximum: (-3, f(-3))
Relative Minimum: (1, f(1))