Question

find all the values of x where the tangent line is horizontal.
$f(x)=2x^{3}+36x^{2}+210x+9$

Explanation:

Step1: Compute derivative of f(x)

$f'(x) = \frac{d}{dx}(2x^3 + 36x^2 + 210x + 9) = 6x^2 + 72x + 210$

Step2: Set derivative to 0 (horizontal tangent)

$6x^2 + 72x + 210 = 0$

Step3: Simplify the equation

Divide by 6: $x^2 + 12x + 35 = 0$

Step4: Factor quadratic equation

$(x + 5)(x + 7) = 0$

Step5: Solve for x

Set each factor to 0: $x + 5 = 0$ or $x + 7 = 0$

Answer:

$x = -7$ and $x = -5$