determine all inflection points of f(x)=x^4 + 2x^3 - 36x^2 - 3x + 4. as your answer, please enter the sum of…

determine all inflection points of f(x)=x^4 + 2x^3 - 36x^2 - 3x + 4. as your answer, please enter the sum of the first coordinates of inflection points that are on the graph of y = f(x).

determine all inflection points of f(x)=x^4 + 2x^3 - 36x^2 - 3x + 4. as your answer, please enter the sum of the first coordinates of inflection points that are on the graph of y = f(x).

Answer

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=4x^{3}+6x^{2}-72x - 3$.

Step2: Find the second - derivative

Differentiate $f'(x)$ again. $f''(x)=12x^{2}+12x - 72$.

Step3: Set the second - derivative equal to zero

$12x^{2}+12x - 72 = 0$. Divide through by 12: $x^{2}+x - 6=0$.

Step4: Solve the quadratic equation

Factor the quadratic equation: $(x + 3)(x - 2)=0$. So $x=-3$ or $x = 2$.

Answer:

$-1$