Question

first, find $f(x)$. then determine all critical values for the function $f(x) = 6x^3 - 18x + 1$, and list them (separated by commas) in the box below. $f(x) = 18x^2 - 18$ list of critical numbers:

Explanation:

Step1: Verify the derivative

Given $f(x)=6x^3-18x+1$, apply power rule $\frac{d}{dx}(x^n)=nx^{n-1}$:
$$f'(x)=6\cdot3x^{2}-18\cdot1+0=18x^2-18$$

Step2: Set derivative to 0

Critical numbers occur where $f'(x)=0$:
$$18x^2-18=0$$

Step3: Solve for $x$

Factor and isolate $x^2$:
$$18(x^2-1)=0 \implies x^2=1$$
Take square roots:
$$x=\pm1$$

Answer:

$-1, 1$