Question

Find the relative extrema of the function and classify each as a maximum or minimum. f(x) = x3 - 12x - 1 Relative maximum: (5, 64); relative minimum: (2, -17) O Relative maximum: (5, 64); relative minimum: (-3, 8) Relative minimum: (-2, 15); relative maximum: (2, -17) Relative maximum: (-2, 15); relative minimum: (2, -17)

Answer 1

To find the relative minimum and maximum, we need to find the derivative for the equation of the function:

$f(x)=x^3-12x-1$

Then, the derivative is (following the rules of derivatives):

$3x^2-12=3(x^2-4)$

To find the relative maximum and minimum, we need to equate the derivative to zero, and solve for x:

$3(x^2-4)=0\Rightarrow x^2-4=0\Rightarrow x^2=4\Rightarrow x=\pm2$

In this case, we have two values for x, x = -2, and x = 2.

Substituting these values in the original equation, we have:

$f(-2)=3\cdot(-2)^2-12(-2)-1\Rightarrow f(-2)=15$

And

$f(2)=3\cdot(2)^2-12\cdot(2)-1\Rightarrow f(2)=-17$

In this case, we have two pair of values:

(-2, 15), and (2, -17).

If we obtain the second derivative for the original value, we have that is 6x.

If we substitute x = -2 in the second derivative, we have 6 * (-2) = -12. Since the value is negative, the point (-2, 15) is a Relative Maximum.

Likewise, if we make x = 2, we have 6 * (2) = 12. Since this value is positive, we have a relative minimum (2, -17).

Therefore:

Relative maximum: (-2, 15).

Relative minimum: (2, -17).

So the correct option is the last one.