Question

let $f(x) = 2x^3 - 15x^2 + 24x - 2$.
find the derivative.
$f(x) =
$the function has has two critical values. the smaller one equals and the larger one equals

Explanation:

Step1: Apply power rule to differentiate

$\frac{d}{dx}[2x^3 -15x^2 +24x -2] = 2*3x^{2} -15*2x^{1} +24*1x^{0} -0$

Step2: Simplify the derivative

$f'(x) = 6x^2 -30x +24$

Step3: Set derivative to 0 for critical values

$6x^2 -30x +24 = 0$

Step4: Divide equation by 6

$x^2 -5x +4 = 0$

Step5: Factor quadratic equation

$(x-1)(x-4) = 0$

Step6: Solve for x values

$x=1$ and $x=4$

Answer:

$f'(x) = 6x^2 - 30x + 24$
The smaller critical value equals $1$ and the larger one equals $4$