Question

find the absolute maximum and absolute minimum values of f on the given interval.

$f(x)=6x^{3}-18x^{2}-54x + 8,-2,4$

step 1

the absolute maximum and minimum values of f occur either at a critical point inside the interval or at an endpoint of the interval. recall that a critical point is a point where $f(x)=0$ or is undefined. we begin by finding the derivative of f.

$f(x)=18x^{2}-36x - 54$

step 2

we now solve $f(x)=0$ for x, which gives the following critical numbers. (enter your answers as a comma - separated list.)

$x=-1,3$

step 3

we must now find the function values at the critical numbers we just found and at the endpoints of the interval $-2,4$.

$f(-1)=$
$f(3)=$
$f(-2)=$
$f(4)=$

Explanation:

Step1: Calculate \(f(-1)\)

Substitute \(x = - 1\) into \(f(x)=6x^{3}-18x^{2}-54x + 8\).

$$
LATEXBLOCK0
$$

Step2: Calculate \(f(3)\)

Substitute \(x = 3\) into \(f(x)=6x^{3}-18x^{2}-54x + 8\).

$$
LATEXBLOCK1
$$

Step3: Calculate \(f(-2)\)

Substitute \(x=-2\) into \(f(x)=6x^{3}-18x^{2}-54x + 8\).

$$
LATEXBLOCK2
$$

Step4: Calculate \(f(4)\)

Substitute \(x = 4\) into \(f(x)=6x^{3}-18x^{2}-54x + 8\).

$$
LATEXBLOCK3
$$

Answer:

\(f(-1)=38\), \(f(3)=-154\), \(f(-2)=-4\), \(f(4)=-112\)