Question

tutorial exercise
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)=

Explanation:

Step1: Differentiate the function

Using the power - rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$, for $f(x)=6x^{3}-18x^{2}-54x + 8$, we have $f'(x)=3\times6x^{2}-2\times18x-54$.
$f'(x)=18x^{2}-36x - 54$

Step2: Simplify the derivative

Factor out 18 from the right - hand side: $f'(x)=18(x^{2}-2x - 3)$

Step3: Further factor the quadratic

Factor the quadratic expression inside the parentheses: $f'(x)=18(x - 3)(x+1)$

Answer:

$18x^{2}-36x - 54$